Primitive. The name Eps_i is a term of type (setprop)set.
Axiom. (Eps_i_ax) We take the following as an axiom:
∀P : setprop, ∀x : set, P xP (Eps_i P)
Definition. We define True to be ∀p : prop, pp of type prop.
Definition. We define False to be ∀p : prop, p of type prop.
Definition. We define not to be λA : propAFalse of type propprop.
Notation. We use ¬ as a prefix operator with priority 700 corresponding to applying term not.
Definition. We define and to be λA B : prop∀p : prop, (ABp)p of type proppropprop.
Notation. We use as an infix operator with priority 780 and which associates to the left corresponding to applying term and.
Definition. We define or to be λA B : prop∀p : prop, (Ap)(Bp)p of type proppropprop.
Notation. We use as an infix operator with priority 785 and which associates to the left corresponding to applying term or.
Definition. We define iff to be λA B : propand (AB) (BA) of type proppropprop.
Notation. We use as an infix operator with priority 805 and no associativity corresponding to applying term iff.
Beginning of Section Eq
Variable A : SType
Definition. We define eq to be λx y : A∀Q : AAprop, Q x yQ y x of type AAprop.
Definition. We define neq to be λx y : A¬ eq x y of type AAprop.
End of Section Eq
Notation. We use = as an infix operator with priority 502 and no associativity corresponding to applying term eq.
Notation. We use as an infix operator with priority 502 and no associativity corresponding to applying term neq.
Beginning of Section FE
Variable A B : SType
Axiom. (func_ext) We take the following as an axiom:
∀f g : AB, (∀x : A, f x = g x)f = g
End of Section FE
Beginning of Section Ex
Variable A : SType
Definition. We define ex to be λQ : Aprop∀P : prop, (∀x : A, Q xP)P of type (Aprop)prop.
End of Section Ex
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex.
Axiom. (prop_ext) We take the following as an axiom:
∀p q : prop, iff p qp = q
Primitive. The name In is a term of type setsetprop.
Notation. We use as an infix operator with priority 500 and no associativity corresponding to applying term In. Furthermore, we may write xA, B to mean x : set, xAB.
Definition. We define Subq to be λA B ⇒ ∀xA, x B of type setsetprop.
Notation. We use as an infix operator with priority 500 and no associativity corresponding to applying term Subq. Furthermore, we may write xA, B to mean x : set, xAB.
Axiom. (set_ext) We take the following as an axiom:
∀X Y : set, X YY XX = Y
Axiom. (In_ind) We take the following as an axiom:
∀P : setprop, (∀X : set, (∀xX, P x)P X)∀X : set, P X
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex and handling ∈ or ⊆ ascriptions using and.
Primitive. The name Empty is a term of type set.
Axiom. (EmptyAx) We take the following as an axiom:
¬ ∃x : set, x Empty
Primitive. The name is a term of type setset.
Axiom. (UnionEq) We take the following as an axiom:
∀X x, x X ∃Y, x Y Y X
Primitive. The name 𝒫 is a term of type setset.
Axiom. (PowerEq) We take the following as an axiom:
∀X Y : set, Y 𝒫 X Y X
Primitive. The name Repl is a term of type set(setset)set.
Notation. {B| xA} is notation for Repl Ax . B).
Axiom. (ReplEq) We take the following as an axiom:
∀A : set, ∀F : setset, ∀y : set, y {F x|xA} ∃xA, y = F x
Definition. We define TransSet to be λU : set∀xU, x U of type setprop.
Definition. We define Union_closed to be λU : set∀X : set, X U X U of type setprop.
Definition. We define Power_closed to be λU : set∀X : set, X U𝒫 X U of type setprop.
Definition. We define Repl_closed to be λU : set∀X : set, X U∀F : setset, (∀x : set, x XF x U){F x|xX} U of type setprop.
Definition. We define ZF_closed to be λU : setUnion_closed U Power_closed U Repl_closed U of type setprop.
Primitive. The name UnivOf is a term of type setset.
Axiom. (UnivOf_In) We take the following as an axiom:
∀N : set, N UnivOf N
Axiom. (UnivOf_TransSet) We take the following as an axiom:
∀N : set, TransSet (UnivOf N)
Axiom. (UnivOf_ZF_closed) We take the following as an axiom:
∀N : set, ZF_closed (UnivOf N)
Axiom. (UnivOf_Min) We take the following as an axiom:
∀N U : set, N UTransSet UZF_closed UUnivOf N U
Theorem. (andI)
∀A B : prop, ABA B
Proof:
An exact proof term for the current goal is (λA B a b P H ⇒ H a b).
Theorem. (orIL)
∀A B : prop, AA B
Proof:
An exact proof term for the current goal is (λA B a P H1 H2 ⇒ H1 a).
Theorem. (orIR)
∀A B : prop, BA B
Proof:
An exact proof term for the current goal is (λA B b P H1 H2 ⇒ H2 b).
Theorem. (iffI)
∀A B : prop, (AB)(BA)(A B)
Proof:
An exact proof term for the current goal is (λA B ⇒ andI (AB) (BA)).
Theorem. (pred_ext)
∀P Q : setprop, (∀x, P x Q x)P = Q
Proof:
Let P and Q be given.
Assume H1.
Apply func_ext set prop to the current goal.
Let x be given.
Apply prop_ext to the current goal.
We will prove P x Q x.
An exact proof term for the current goal is H1 x.
Definition. We define nIn to be λx X ⇒ ¬ In x X of type setsetprop.
Notation. We use as an infix operator with priority 502 and no associativity corresponding to applying term nIn.
Theorem. (EmptyE)
∀x : set, x Empty
Proof:
Let x be given.
Assume H.
Apply EmptyAx to the current goal.
We use x to witness the existential quantifier.
An exact proof term for the current goal is H.
Theorem. (PowerI)
∀X Y : set, Y XY 𝒫 X
Proof:
Let X and Y be given.
Apply PowerEq X Y to the current goal.
An exact proof term for the current goal is (λ_ H ⇒ H).
Theorem. (Subq_Empty)
∀X : set, Empty X
Proof:
An exact proof term for the current goal is (λ(X x : set)(H : x Empty) ⇒ EmptyE x H (x X)).
Theorem. (Empty_In_Power)
∀X : set, Empty 𝒫 X
Proof:
An exact proof term for the current goal is (λX : setPowerI X Empty (Subq_Empty X)).
Theorem. (xm)
∀P : prop, P ¬ P
Proof:
Let P of type prop be given.
Set p1 to be the term λx : setx = Empty P.
Set p2 to be the term λx : setx Empty P.
We prove the intermediate claim L1: p1 Empty.
We will prove (Empty = Empty P).
Apply orIL to the current goal.
An exact proof term for the current goal is (λq H ⇒ H).
We prove the intermediate claim L2: (Eps_i p1) = Empty P.
An exact proof term for the current goal is (Eps_i_ax p1 Empty L1).
We prove the intermediate claim L3: p2 (𝒫 Empty).
We will prove ¬ (𝒫 Empty = Empty) P.
Apply orIL to the current goal.
Assume H1: 𝒫 Empty = Empty.
Apply EmptyE Empty to the current goal.
We will prove Empty Empty.
rewrite the current goal using H1 (from right to left) at position 2.
Apply Empty_In_Power to the current goal.
We prove the intermediate claim L4: Eps_i p2 Empty P.
An exact proof term for the current goal is (Eps_i_ax p2 (𝒫 Empty) L3).
Apply L2 to the current goal.
Assume H1: Eps_i p1 = Empty.
Apply L4 to the current goal.
Assume H2: Eps_i p2 Empty.
We will prove P ¬ P.
Apply orIR to the current goal.
We will prove ¬ P.
Assume H3: P.
We prove the intermediate claim L5: p1 = p2.
Apply pred_ext to the current goal.
Let x be given.
Apply iffI to the current goal.
Assume H4.
We will prove (¬ (x = Empty) P).
Apply orIR to the current goal.
We will prove P.
An exact proof term for the current goal is H3.
Assume H4.
We will prove (x = Empty P).
Apply orIR to the current goal.
We will prove P.
An exact proof term for the current goal is H3.
Apply H2 to the current goal.
rewrite the current goal using L5 (from right to left).
An exact proof term for the current goal is H1.
Assume H2: P.
We will prove P ¬ P.
Apply orIL to the current goal.
We will prove P.
An exact proof term for the current goal is H2.
Assume H1: P.
We will prove P ¬ P.
Apply orIL to the current goal.
We will prove P.
An exact proof term for the current goal is H1.
Theorem. (FalseE)
False∀p : prop, p
Proof:
An exact proof term for the current goal is (λH ⇒ H).
Theorem. (andEL)
∀A B : prop, A BA
Proof:
An exact proof term for the current goal is (λA B H ⇒ H A (λa b ⇒ a)).
Theorem. (andER)
∀A B : prop, A BB
Proof:
An exact proof term for the current goal is (λA B H ⇒ H B (λa b ⇒ b)).
Beginning of Section PropN
Variable P1 P2 P3 : prop
Theorem. (and3I)
P1P2P3P1 P2 P3
Proof:
An exact proof term for the current goal is (λH1 H2 H3 ⇒ andI (P1 P2) P3 (andI P1 P2 H1 H2) H3).
Theorem. (and3E)
P1 P2 P3(∀p : prop, (P1P2P3p)p)
Proof:
An exact proof term for the current goal is (λu p H ⇒ u p (λu u3 ⇒ u p (λu1 u2 ⇒ H u1 u2 u3))).
Theorem. (or3I1)
P1P1 P2 P3
Proof:
An exact proof term for the current goal is (λu ⇒ orIL (P1 P2) P3 (orIL P1 P2 u)).
Theorem. (or3I2)
P2P1 P2 P3
Proof:
An exact proof term for the current goal is (λu ⇒ orIL (P1 P2) P3 (orIR P1 P2 u)).
Theorem. (or3I3)
P3P1 P2 P3
Proof:
An exact proof term for the current goal is (orIR (P1 P2) P3).
Theorem. (or3E)
P1 P2 P3(∀p : prop, (P1p)(P2p)(P3p)p)
Proof:
An exact proof term for the current goal is (λu p H1 H2 H3 ⇒ u p (λu ⇒ u p H1 H2) H3).
Variable P4 : prop
Theorem. (and4I)
P1P2P3P4P1 P2 P3 P4
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 ⇒ andI (P1 P2 P3) P4 (and3I H1 H2 H3) H4).
Variable P5 : prop
Theorem. (and5I)
P1P2P3P4P5P1 P2 P3 P4 P5
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 ⇒ andI (P1 P2 P3 P4) P5 (and4I H1 H2 H3 H4) H5).
Variable P6 : prop
Theorem. (and6I)
P1P2P3P4P5P6P1 P2 P3 P4 P5 P6
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 H6 ⇒ andI (P1 P2 P3 P4 P5) P6 (and5I H1 H2 H3 H4 H5) H6).
Variable P7 : prop
Theorem. (and7I)
P1P2P3P4P5P6P7P1 P2 P3 P4 P5 P6 P7
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 H6 H7 ⇒ andI (P1 P2 P3 P4 P5 P6) P7 (and6I H1 H2 H3 H4 H5 H6) H7).
End of Section PropN
Theorem. (not_or_and_demorgan)
∀A B : prop, ¬ (A B)¬ A ¬ B
Proof:
Let A and B be given.
Assume u: ¬ (A B).
Apply andI to the current goal.
We will prove ¬ A.
Assume a: A.
An exact proof term for the current goal is (u (orIL A B a)).
We will prove ¬ B.
Assume b: B.
An exact proof term for the current goal is (u (orIR A B b)).
Theorem. (not_ex_all_demorgan_i)
∀P : setprop, (¬ ∃x, P x)∀x, ¬ P x
Proof:
Let P be given.
Assume H1.
Let x be given.
Assume H2.
Apply H1 to the current goal.
We use x to witness the existential quantifier.
An exact proof term for the current goal is H2.
Theorem. (iffEL)
∀A B : prop, (A B)AB
Proof:
An exact proof term for the current goal is (λA B ⇒ andEL (AB) (BA)).
Theorem. (iffER)
∀A B : prop, (A B)BA
Proof:
An exact proof term for the current goal is (λA B ⇒ andER (AB) (BA)).
Theorem. (iff_refl)
∀A : prop, A A
Proof:
An exact proof term for the current goal is (λA : propandI (AA) (AA) (λH : AH) (λH : AH)).
Theorem. (iff_sym)
∀A B : prop, (A B)(B A)
Proof:
Let A and B be given.
Assume H1: (AB) (BA).
Apply H1 to the current goal.
Assume H2: AB.
Assume H3: BA.
An exact proof term for the current goal is iffI B A H3 H2.
Theorem. (iff_trans)
∀A B C : prop, (A B)(B C)(A C)
Proof:
Let A, B and C be given.
Assume H1: A B.
Assume H2: B C.
Apply H1 to the current goal.
Assume H3: AB.
Assume H4: BA.
Apply H2 to the current goal.
Assume H5: BC.
Assume H6: CB.
An exact proof term for the current goal is (iffI A C (λH ⇒ H5 (H3 H)) (λH ⇒ H4 (H6 H))).
Theorem. (eq_i_tra)
∀x y z, x = yy = zx = z
Proof:
Let x, y and z be given.
Assume H1 H2.
rewrite the current goal using H2 (from right to left).
An exact proof term for the current goal is H1.
Theorem. (neq_i_sym)
∀x y, x yy x
Proof:
Let x and y be given.
Assume H1 H2.
Apply H1 to the current goal.
Use symmetry.
An exact proof term for the current goal is H2.
Theorem. (Eps_i_ex)
∀P : setprop, (∃x, P x)P (Eps_i P)
Proof:
Let P be given.
Assume H1.
Apply H1 to the current goal.
Let x be given.
Assume H2.
An exact proof term for the current goal is Eps_i_ax P x H2.
Theorem. (prop_ext_2)
∀p q : prop, (pq)(qp)p = q
Proof:
Let p and q be given.
Assume H1 H2.
Apply prop_ext to the current goal.
Apply iffI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Theorem. (Subq_ref)
∀X : set, X X
Proof:
An exact proof term for the current goal is (λ(X x : set)(H : x X) ⇒ H).
Theorem. (Subq_tra)
∀X Y Z : set, X YY ZX Z
Proof:
An exact proof term for the current goal is (λ(X Y Z : set)(H1 : X Y)(H2 : Y Z)(x : set)(H : x X) ⇒ (H2 x (H1 x H))).
Theorem. (Empty_Subq_eq)
∀X : set, X EmptyX = Empty
Proof:
Let X be given.
Assume H1: X Empty.
Apply set_ext to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is (Subq_Empty X).
Theorem. (Empty_eq)
∀X : set, (∀x, x X)X = Empty
Proof:
Let X be given.
Assume H1: ∀x, x X.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume H2: x X.
We will prove False.
An exact proof term for the current goal is (H1 x H2).
Theorem. (UnionI)
∀X x Y : set, x YY Xx X
Proof:
Let X, x and Y be given.
Assume H1: x Y.
Assume H2: Y X.
Apply UnionEq X x to the current goal.
Assume _ H3.
Apply H3 to the current goal.
We will prove ∃Y : set, x Y Y X.
We use Y to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Theorem. (UnionE)
∀X x : set, x X∃Y : set, x Y Y X
Proof:
An exact proof term for the current goal is (λX x : setiffEL (x X) (∃Y : set, x Y Y X) (UnionEq X x)).
Theorem. (UnionE_impred)
∀X x : set, x X∀p : prop, (∀Y : set, x YY Xp)p
Proof:
Let X and x be given.
Assume H1.
Let p be given.
Assume Hp.
Apply UnionE X x H1 to the current goal.
Let x be given.
Assume H2.
Apply H2 to the current goal.
An exact proof term for the current goal is Hp x.
Theorem. (PowerE)
∀X Y : set, Y 𝒫 XY X
Proof:
Let X and Y be given.
Apply PowerEq X Y to the current goal.
An exact proof term for the current goal is (λH _ ⇒ H).
Theorem. (Self_In_Power)
∀X : set, X 𝒫 X
Proof:
An exact proof term for the current goal is (λX : setPowerI X X (Subq_ref X)).
Theorem. (dneg)
∀P : prop, ¬ ¬ PP
Proof:
Let P be given.
Assume H1.
Apply xm P to the current goal.
An exact proof term for the current goal is (λH ⇒ H).
Assume H2: ¬ P.
We will prove False.
An exact proof term for the current goal is H1 H2.
Theorem. (not_all_ex_demorgan_i)
∀P : setprop, ¬ (∀x, P x)∃x, ¬ P x
Proof:
Let P be given.
Assume u: ¬ ∀x, P x.
Apply dneg to the current goal.
Assume v: ¬ ∃x, ¬ P x.
Apply u to the current goal.
Let x be given.
Apply dneg to the current goal.
Assume w: ¬ P x.
An exact proof term for the current goal is (not_ex_all_demorgan_i (λx ⇒ ¬ P x) v x w).
Theorem. (eq_or_nand)
or = (λx y : prop¬ (¬ x ¬ y))
Proof:
Apply func_ext prop (propprop) to the current goal.
Let x be given.
Apply func_ext prop prop to the current goal.
Let y be given.
Apply prop_ext_2 to the current goal.
Assume H1: x y.
Assume H2: ¬ x ¬ y.
Apply H2 to the current goal.
Assume H3 H4.
An exact proof term for the current goal is (H1 False H3 H4).
Assume H1: ¬ (¬ x ¬ y).
Apply (xm x) to the current goal.
Assume H2: x.
Apply orIL to the current goal.
An exact proof term for the current goal is H2.
Assume H2: ¬ x.
Apply (xm y) to the current goal.
Assume H3: y.
Apply orIR to the current goal.
An exact proof term for the current goal is H3.
Assume H3: ¬ y.
Apply H1 to the current goal.
An exact proof term for the current goal is (andI (¬ x) (¬ y) H2 H3).
Definition. We define exactly1of2 to be λA B : propA ¬ B ¬ A B of type proppropprop.
Theorem. (exactly1of2_I1)
∀A B : prop, A¬ Bexactly1of2 A B
Proof:
Let A and B be given.
Assume HA: A.
Assume HB: ¬ B.
We will prove A ¬ B ¬ A B.
Apply orIL to the current goal.
We will prove A ¬ B.
An exact proof term for the current goal is (andI A (¬ B) HA HB).
Theorem. (exactly1of2_I2)
∀A B : prop, ¬ ABexactly1of2 A B
Proof:
Let A and B be given.
Assume HA: ¬ A.
Assume HB: B.
We will prove A ¬ B ¬ A B.
Apply orIR to the current goal.
We will prove ¬ A B.
An exact proof term for the current goal is (andI (¬ A) B HA HB).
Theorem. (exactly1of2_E)
∀A B : prop, exactly1of2 A B∀p : prop, (A¬ Bp)(¬ ABp)p
Proof:
Let A and B be given.
Assume H1: exactly1of2 A B.
Let p be given.
Assume H2: A¬ Bp.
Assume H3: ¬ ABp.
Apply (H1 p) to the current goal.
An exact proof term for the current goal is (λH4 : A ¬ BH4 p H2).
An exact proof term for the current goal is (λH4 : ¬ A BH4 p H3).
Theorem. (exactly1of2_or)
∀A B : prop, exactly1of2 A BA B
Proof:
Let A and B be given.
Assume H1: exactly1of2 A B.
Apply (exactly1of2_E A B H1 (A B)) to the current goal.
An exact proof term for the current goal is (λ(HA : A)(_ : ¬ B) ⇒ orIL A B HA).
An exact proof term for the current goal is (λ(_ : ¬ A)(HB : B) ⇒ orIR A B HB).
Theorem. (ReplI)
∀A : set, ∀F : setset, ∀x : set, x AF x {F x|xA}
Proof:
Let A, F and x be given.
Assume H1.
Apply ReplEq A F (F x) to the current goal.
Assume _ H2.
Apply H2 to the current goal.
We will prove ∃x'A, F x = F x'.
We use x to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is (λq H ⇒ H).
Theorem. (ReplE)
∀A : set, ∀F : setset, ∀y : set, y {F x|xA}∃xA, y = F x
Proof:
Let A, F and y be given.
Apply ReplEq A F y to the current goal.
An exact proof term for the current goal is (λH _ ⇒ H).
Theorem. (ReplE_impred)
∀A : set, ∀F : setset, ∀y : set, y {F x|xA}∀p : prop, (∀x : set, x Ay = F xp)p
Proof:
Let A, F and y be given.
Assume H1.
Apply ReplE A F y H1 to the current goal.
Let x be given.
Assume H2.
Apply H2 to the current goal.
Assume H3 H4.
Let p be given.
Assume Hp.
An exact proof term for the current goal is Hp x H3 H4.
Theorem. (ReplE')
∀X, ∀f : setset, ∀p : setprop, (∀xX, p (f x))∀y{f x|xX}, p y
Proof:
Let X, f and p be given.
Assume H1.
Let y be given.
Assume Hy.
Apply ReplE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx: x X.
Assume Hx2: y = f x.
We will prove p y.
rewrite the current goal using Hx2 (from left to right).
An exact proof term for the current goal is H1 x Hx.
Theorem. (Repl_Empty)
∀F : setset, {F x|xEmpty} = Empty
Proof:
Let F be given.
Apply (Empty_eq {F x|xEmpty}) to the current goal.
Let y be given.
Assume H1: y {F x|xEmpty}.
Apply (ReplE_impred Empty F y H1) to the current goal.
Let x be given.
Assume H2: x Empty.
Assume _.
An exact proof term for the current goal is (EmptyE x H2).
Theorem. (ReplEq_ext_sub)
∀X, ∀F G : setset, (∀xX, F x = G x){F x|xX} {G x|xX}
Proof:
Let X, F and G be given.
Assume H1: ∀xX, F x = G x.
Let y be given.
Assume Hy: y {F x|xX}.
Apply ReplE_impred X F y Hy to the current goal.
Let x be given.
Assume Hx: x X.
Assume H2: y = F x.
We will prove y {G x|xX}.
rewrite the current goal using H2 (from left to right).
We will prove F x {G x|xX}.
rewrite the current goal using H1 x Hx (from left to right).
We will prove G x {G x|xX}.
Apply ReplI to the current goal.
An exact proof term for the current goal is Hx.
Theorem. (ReplEq_ext)
∀X, ∀F G : setset, (∀xX, F x = G x){F x|xX} = {G x|xX}
Proof:
Let X, F and G be given.
Assume H1: ∀xX, F x = G x.
Apply set_ext to the current goal.
An exact proof term for the current goal is ReplEq_ext_sub X F G H1.
Apply ReplEq_ext_sub X G F to the current goal.
Let x be given.
Assume Hx.
Use symmetry.
An exact proof term for the current goal is H1 x Hx.
Theorem. (Repl_inv_eq)
∀P : setprop, ∀f g : setset, (∀x, P xg (f x) = x)∀X, (∀xX, P x){g y|y{f x|xX}} = X
Proof:
Let P, f and g be given.
Assume H1.
Let X be given.
Assume HX.
Apply set_ext to the current goal.
Let w be given.
Assume Hw: w {g y|y{f x|xX}}.
Apply ReplE_impred {f x|xX} g w Hw to the current goal.
Let y be given.
Assume Hy: y {f x|xX}.
Assume Hwy: w = g y.
Apply ReplE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx: x X.
Assume Hyx: y = f x.
We will prove w X.
rewrite the current goal using Hwy (from left to right).
rewrite the current goal using Hyx (from left to right).
We will prove g (f x) X.
rewrite the current goal using H1 x (HX x Hx) (from left to right).
An exact proof term for the current goal is Hx.
Let x be given.
Assume Hx: x X.
rewrite the current goal using H1 x (HX x Hx) (from right to left).
We will prove g (f x) {g y|y{f x|xX}}.
Apply ReplI to the current goal.
We will prove f x {f x|xX}.
Apply ReplI to the current goal.
An exact proof term for the current goal is Hx.
Theorem. (Repl_invol_eq)
∀P : setprop, ∀f : setset, (∀x, P xf (f x) = x)∀X, (∀xX, P x){f y|y{f x|xX}} = X
Proof:
Let P and f be given.
Assume H1.
An exact proof term for the current goal is Repl_inv_eq P f f H1.
Definition. We define If_i to be (λp x y ⇒ Eps_i (λz : setp z = x ¬ p z = y)) of type propsetsetset.
Notation. if cond then T else E is notation corresponding to If_i type cond T E where type is the inferred type of T.
Theorem. (If_i_correct)
∀p : prop, ∀x y : set, p (if p then x else y) = x ¬ p (if p then x else y) = y
Proof:
Let p, x and y be given.
Apply (xm p) to the current goal.
Assume H1: p.
We prove the intermediate claim L1: p x = x ¬ p x = y.
Apply orIL to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Use reflexivity.
An exact proof term for the current goal is (Eps_i_ax (λz : setp z = x ¬ p z = y) x L1).
Assume H1: ¬ p.
We prove the intermediate claim L1: p y = x ¬ p y = y.
Apply orIR to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Use reflexivity.
An exact proof term for the current goal is (Eps_i_ax (λz : setp z = x ¬ p z = y) y L1).
Theorem. (If_i_0)
∀p : prop, ∀x y : set, ¬ p(if p then x else y) = y
Proof:
Let p, x and y be given.
Assume H1: ¬ p.
Apply (If_i_correct p x y) to the current goal.
Assume H2: p (if p then x else y) = x.
An exact proof term for the current goal is (H1 (andEL p ((if p then x else y) = x) H2) ((if p then x else y) = y)).
Assume H2: ¬ p (if p then x else y) = y.
An exact proof term for the current goal is (andER (¬ p) ((if p then x else y) = y) H2).
Theorem. (If_i_1)
∀p : prop, ∀x y : set, p(if p then x else y) = x
Proof:
Let p, x and y be given.
Assume H1: p.
Apply (If_i_correct p x y) to the current goal.
Assume H2: p (if p then x else y) = x.
An exact proof term for the current goal is (andER p ((if p then x else y) = x) H2).
Assume H2: ¬ p (if p then x else y) = y.
An exact proof term for the current goal is (andEL (¬ p) ((if p then x else y) = y) H2 H1 ((if p then x else y) = x)).
Theorem. (If_i_or)
∀p : prop, ∀x y : set, (if p then x else y) = x (if p then x else y) = y
Proof:
Let p, x and y be given.
Apply (xm p) to the current goal.
Assume H1: p.
Apply orIL to the current goal.
An exact proof term for the current goal is (If_i_1 p x y H1).
Assume H1: ¬ p.
Apply orIR to the current goal.
An exact proof term for the current goal is (If_i_0 p x y H1).
Definition. We define UPair to be λy z ⇒ {if Empty X then y else z|X𝒫 (𝒫 Empty)} of type setsetset.
Notation. {x,y} is notation for UPair x y.
Theorem. (UPairE)
∀x y z : set, x {y,z}x = y x = z
Proof:
Let x, y and z be given.
Assume H1: x {y,z}.
Apply (ReplE (𝒫 (𝒫 Empty)) (λX ⇒ if Empty X then y else z) x H1) to the current goal.
Let X be given.
Assume H2: X 𝒫 (𝒫 Empty) x = if Empty X then y else z.
We prove the intermediate claim L1: x = if Empty X then y else z.
An exact proof term for the current goal is (andER (X 𝒫 (𝒫 Empty)) (x = if Empty X then y else z) H2).
Apply (If_i_or (Empty X) y z) to the current goal.
Assume H3: (if Empty X then y else z) = y.
Apply orIL to the current goal.
We will prove x = y.
rewrite the current goal using H3 (from right to left).
An exact proof term for the current goal is L1.
Assume H3: (if Empty X then y else z) = z.
Apply orIR to the current goal.
We will prove x = z.
rewrite the current goal using H3 (from right to left).
An exact proof term for the current goal is L1.
Theorem. (UPairI1)
∀y z : set, y {y,z}
Proof:
Let y and z be given.
We will prove y {y,z}.
rewrite the current goal using (If_i_1 (Empty 𝒫 Empty) y z (Empty_In_Power Empty)) (from right to left) at position 1.
We will prove (if Empty 𝒫 Empty then y else z) {y,z}.
Apply (ReplI (𝒫 (𝒫 Empty)) (λX : setif (Empty X) then y else z) (𝒫 Empty)) to the current goal.
We will prove 𝒫 Empty 𝒫 (𝒫 Empty).
An exact proof term for the current goal is (Self_In_Power (𝒫 Empty)).
Theorem. (UPairI2)
∀y z : set, z {y,z}
Proof:
Let y and z be given.
We will prove z {y,z}.
rewrite the current goal using (If_i_0 (Empty Empty) y z (EmptyE Empty)) (from right to left) at position 1.
We will prove (if Empty Empty then y else z) {y,z}.
Apply (ReplI (𝒫 (𝒫 Empty)) (λX : setif (Empty X) then y else z) Empty) to the current goal.
We will prove Empty 𝒫 (𝒫 Empty).
An exact proof term for the current goal is (Empty_In_Power (𝒫 Empty)).
Definition. We define Sing to be λx ⇒ {x,x} of type setset.
Notation. {x} is notation for Sing x.
Theorem. (SingI)
∀x : set, x {x}
Proof:
An exact proof term for the current goal is (λx : setUPairI1 x x).
Theorem. (SingE)
∀x y : set, y {x}y = x
Proof:
An exact proof term for the current goal is (λx y H ⇒ UPairE y x x H (y = x) (λH ⇒ H) (λH ⇒ H)).
Definition. We define binunion to be λX Y ⇒ {X,Y} of type setsetset.
Notation. We use as an infix operator with priority 345 and which associates to the left corresponding to applying term binunion.
Theorem. (binunionI1)
∀X Y z : set, z Xz X Y
Proof:
Let X, Y and z be given.
Assume H1: z X.
We will prove z X Y.
We will prove z {X,Y}.
Apply (UnionI {X,Y} z X) to the current goal.
We will prove z X.
An exact proof term for the current goal is H1.
We will prove X {X,Y}.
An exact proof term for the current goal is (UPairI1 X Y).
Theorem. (binunionI2)
∀X Y z : set, z Yz X Y
Proof:
Let X, Y and z be given.
Assume H1: z Y.
We will prove z X Y.
We will prove z {X,Y}.
Apply (UnionI {X,Y} z Y) to the current goal.
We will prove z Y.
An exact proof term for the current goal is H1.
We will prove Y {X,Y}.
An exact proof term for the current goal is (UPairI2 X Y).
Theorem. (binunionE)
∀X Y z : set, z X Yz X z Y
Proof:
Let X, Y and z be given.
Assume H1: z X Y.
We will prove z X z Y.
Apply (UnionE_impred {X,Y} z H1) to the current goal.
Let Z be given.
Assume H2: z Z.
Assume H3: Z {X,Y}.
Apply (UPairE Z X Y H3) to the current goal.
Assume H4: Z = X.
Apply orIL to the current goal.
We will prove z X.
rewrite the current goal using H4 (from right to left).
We will prove z Z.
An exact proof term for the current goal is H2.
Assume H4: Z = Y.
Apply orIR to the current goal.
We will prove z Y.
rewrite the current goal using H4 (from right to left).
We will prove z Z.
An exact proof term for the current goal is H2.
Theorem. (binunionE')
∀X Y z, ∀p : prop, (z Xp)(z Yp)(z X Yp)
Proof:
Let X, Y, z and p be given.
Assume H1 H2 Hz.
Apply binunionE X Y z Hz to the current goal.
Assume H3: z X.
An exact proof term for the current goal is H1 H3.
Assume H3: z Y.
An exact proof term for the current goal is H2 H3.
Theorem. (binunion_asso)
∀X Y Z : set, X (Y Z) = (X Y) Z
Proof:
Let X, Y and Z be given.
Apply set_ext to the current goal.
Let w be given.
Assume H1: w X (Y Z).
We will prove w (X Y) Z.
Apply (binunionE X (Y Z) w H1) to the current goal.
Assume H2: w X.
Apply binunionI1 to the current goal.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H2.
Assume H2: w Y Z.
Apply (binunionE Y Z w H2) to the current goal.
Assume H3: w Y.
Apply binunionI1 to the current goal.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H3.
Assume H3: w Z.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H3.
Let w be given.
Assume H1: w (X Y) Z.
We will prove w X (Y Z).
Apply (binunionE (X Y) Z w H1) to the current goal.
Assume H2: w X Y.
Apply (binunionE X Y w H2) to the current goal.
Assume H3: w X.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H3.
Assume H3: w Y.
Apply binunionI2 to the current goal.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H3.
Assume H2: w Z.
Apply binunionI2 to the current goal.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
Theorem. (binunion_com_Subq)
∀X Y : set, X Y Y X
Proof:
Let X, Y and w be given.
Assume H1: w X Y.
We will prove w Y X.
Apply (binunionE X Y w H1) to the current goal.
Assume H2: w X.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
Assume H2: w Y.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H2.
Theorem. (binunion_com)
∀X Y : set, X Y = Y X
Proof:
Let X and Y be given.
Apply set_ext to the current goal.
An exact proof term for the current goal is (binunion_com_Subq X Y).
An exact proof term for the current goal is (binunion_com_Subq Y X).
Theorem. (binunion_idl)
∀X : set, Empty X = X
Proof:
Let X be given.
Apply set_ext to the current goal.
Let x be given.
Assume H1: x Empty X.
Apply (binunionE Empty X x H1) to the current goal.
Assume H2: x Empty.
We will prove False.
An exact proof term for the current goal is (EmptyE x H2).
Assume H2: x X.
An exact proof term for the current goal is H2.
Let x be given.
Assume H2: x X.
We will prove x Empty X.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
Theorem. (binunion_idr)
∀X : set, X Empty = X
Proof:
Let X be given.
rewrite the current goal using (binunion_com X Empty) (from left to right).
An exact proof term for the current goal is (binunion_idl X).
Theorem. (binunion_Subq_1)
∀X Y : set, X X Y
Proof:
An exact proof term for the current goal is binunionI1.
Theorem. (binunion_Subq_2)
∀X Y : set, Y X Y
Proof:
An exact proof term for the current goal is binunionI2.
Theorem. (binunion_Subq_min)
∀X Y Z : set, X ZY ZX Y Z
Proof:
Let X, Y and Z be given.
Assume H1: X Z.
Assume H2: Y Z.
Let w be given.
Assume H3: w X Y.
Apply (binunionE X Y w H3) to the current goal.
Assume H4: w X.
An exact proof term for the current goal is (H1 w H4).
Assume H4: w Y.
An exact proof term for the current goal is (H2 w H4).
Theorem. (Subq_binunion_eq)
∀X Y, (X Y) = (X Y = Y)
Proof:
Let X and Y be given.
Apply prop_ext_2 to the current goal.
Assume H1: X Y.
We will prove X Y = Y.
Apply set_ext to the current goal.
We will prove X Y Y.
Apply (binunion_Subq_min X Y Y) to the current goal.
We will prove X Y.
An exact proof term for the current goal is H1.
We will prove Y Y.
An exact proof term for the current goal is (Subq_ref Y).
We will prove Y X Y.
An exact proof term for the current goal is (binunion_Subq_2 X Y).
Assume H1: X Y = Y.
We will prove X Y.
rewrite the current goal using H1 (from right to left).
We will prove X X Y.
An exact proof term for the current goal is (binunion_Subq_1 X Y).
Definition. We define SetAdjoin to be λX y ⇒ X {y} of type setsetset.
Notation. We now use the set enumeration notation {...,...,...} in general. If 0 elements are given, then Empty is used to form the corresponding term. If 1 element is given, then Sing is used to form the corresponding term. If 2 elements are given, then UPair is used to form the corresponding term. If more than elements are given, then SetAdjoin is used to reduce to the case with one fewer elements.
Definition. We define famunion to be λX F ⇒ {F x|xX} of type set(setset)set.
Notation. We use x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using famunion.
Theorem. (famunionI)
∀X : set, ∀F : (setset), ∀x y : set, x Xy F xy xXF x
Proof:
An exact proof term for the current goal is (λX F x y H1 H2 ⇒ UnionI (Repl X F) y (F x) H2 (ReplI X F x H1)).
Theorem. (famunionE)
∀X : set, ∀F : (setset), ∀y : set, y (xXF x)∃xX, y F x
Proof:
Let X, F and y be given.
Assume H1: y (xXF x).
We will prove ∃xX, y F x.
Apply (UnionE_impred {F x|xX} y H1) to the current goal.
Let Y be given.
Assume H2: y Y.
Assume H3: Y {F x|xX}.
Apply (ReplE_impred X F Y H3) to the current goal.
Let x be given.
Assume H4: x X.
Assume H5: Y = F x.
We use x to witness the existential quantifier.
We will prove x X y F x.
Apply andI to the current goal.
An exact proof term for the current goal is H4.
We will prove y F x.
rewrite the current goal using H5 (from right to left).
An exact proof term for the current goal is H2.
Theorem. (famunionE_impred)
∀X : set, ∀F : (setset), ∀y : set, y (xXF x)∀p : prop, (∀x, x Xy F xp)p
Proof:
Let X, F and y be given.
Assume Hy.
Let p be given.
Assume Hp.
Apply famunionE X F y Hy to the current goal.
Let x be given.
Assume H1.
Apply H1 to the current goal.
An exact proof term for the current goal is Hp x.
Theorem. (famunion_Empty)
∀F : setset, (xEmptyF x) = Empty
Proof:
Let F be given.
Apply Empty_Subq_eq to the current goal.
Let y be given.
Assume Hy: y xEmptyF x.
Apply famunionE_impred Empty F y Hy to the current goal.
Let x be given.
Assume Hx: x Empty.
We will prove False.
An exact proof term for the current goal is EmptyE x Hx.
Theorem. (famunion_Subq)
∀X, ∀f g : setset, (∀xX, f x g x)famunion X f famunion X g
Proof:
Let X, f and g be given.
Assume Hfg.
Let y be given.
Assume Hy.
Apply famunionE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx.
Assume H1: y f x.
Apply famunionI X g x y Hx to the current goal.
We will prove y g x.
An exact proof term for the current goal is Hfg x Hx y H1.
Theorem. (famunion_ext)
∀X, ∀f g : setset, (∀xX, f x = g x)famunion X f = famunion X g
Proof:
Let X, f and g be given.
Assume Hfg.
Apply set_ext to the current goal.
Apply famunion_Subq to the current goal.
Let x be given.
Assume Hx.
rewrite the current goal using Hfg x Hx (from left to right).
Apply Subq_ref to the current goal.
Apply famunion_Subq to the current goal.
Let x be given.
Assume Hx.
rewrite the current goal using Hfg x Hx (from left to right).
Apply Subq_ref to the current goal.
Beginning of Section SepSec
Variable X : set
Variable P : setprop
Let z : setEps_i (λz ⇒ z X P z)
Let F : setsetλx ⇒ if P x then x else z
Definition. We define Sep to be if (∃zX, P z) then {F x|xX} else Empty of type set.
End of Section SepSec
Notation. {xA | B} is notation for Sep Ax . B).
Theorem. (SepI)
∀X : set, ∀P : (setprop), ∀x : set, x XP xx {xX|P x}
Proof:
Let X, P and x be given.
Set z to be the term Eps_i (λz ⇒ z X P z) of type set.
Set F to be the term λx ⇒ if P x then x else z of type setset.
Assume H1: x X.
Assume H2: P x.
We prove the intermediate claim L1: ∃zX, P z.
We use x to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
We will prove x {xX|P x}.
We will prove x if (∃zX, P z) then {F x|xX} else Empty.
We prove the intermediate claim L2: (if (∃zX, P z) then {F x|xX} else Empty) = {F x|xX}.
An exact proof term for the current goal is (If_i_1 (∃zX, P z) {F x|xX} Empty L1).
rewrite the current goal using L2 (from left to right).
We will prove x {F x|xX}.
We prove the intermediate claim L3: F x = x.
We will prove (if P x then x else z) = x.
An exact proof term for the current goal is (If_i_1 (P x) x z H2).
rewrite the current goal using L3 (from right to left).
We will prove F x {F x|xX}.
An exact proof term for the current goal is (ReplI X F x H1).
Theorem. (SepE)
∀X : set, ∀P : (setprop), ∀x : set, x {xX|P x}x X P x
Proof:
Let X, P and x be given.
Set z to be the term Eps_i (λz ⇒ z X P z) of type set.
Set F to be the term λx ⇒ if P x then x else z of type setset.
Apply (xm (∃zX, P z)) to the current goal.
Assume H1: ∃zX, P z.
We will prove (x (if (∃zX, P z) then {F x|xX} else Empty)x X P x).
We prove the intermediate claim L1: (if (∃zX, P z) then {F x|xX} else Empty) = {F x|xX}.
An exact proof term for the current goal is (If_i_1 (∃zX, P z) {F x|xX} Empty H1).
rewrite the current goal using L1 (from left to right).
We will prove x {F x|xX}x X P x.
Assume H2: x {F x|xX}.
Apply (ReplE_impred X F x H2) to the current goal.
Let y be given.
Assume H3: y X.
Assume H4: x = F y.
We will prove x X P x.
Apply (xm (P y)) to the current goal.
Assume H5: P y.
We prove the intermediate claim L2: x = y.
rewrite the current goal using (If_i_1 (P y) y z H5) (from right to left).
An exact proof term for the current goal is H4.
rewrite the current goal using L2 (from left to right).
We will prove y X P y.
Apply andI to the current goal.
An exact proof term for the current goal is H3.
An exact proof term for the current goal is H5.
Assume H5: ¬ P y.
We prove the intermediate claim L2: x = z.
rewrite the current goal using (If_i_0 (P y) y z H5) (from right to left).
An exact proof term for the current goal is H4.
rewrite the current goal using L2 (from left to right).
We will prove z X P z.
An exact proof term for the current goal is (Eps_i_ex (λz ⇒ z X P z) H1).
Assume H1: ¬ ∃zX, P z.
We will prove (x (if (∃zX, P z) then {F x|xX} else Empty)x X P x).
We prove the intermediate claim L1: (if (∃zX, P z) then {F x|xX} else Empty) = Empty.
An exact proof term for the current goal is (If_i_0 (∃zX, P z) {F x|xX} Empty H1).
rewrite the current goal using L1 (from left to right).
We will prove x Emptyx X P x.
Assume H2: x Empty.
We will prove False.
An exact proof term for the current goal is (EmptyE x H2).
Theorem. (SepE1)
∀X : set, ∀P : (setprop), ∀x : set, x {xX|P x}x X
Proof:
An exact proof term for the current goal is (λX P x H ⇒ SepE X P x H (x X) (λH _ ⇒ H)).
Theorem. (SepE2)
∀X : set, ∀P : (setprop), ∀x : set, x {xX|P x}P x
Proof:
An exact proof term for the current goal is (λX P x H ⇒ SepE X P x H (P x) (λ_ H ⇒ H)).
Theorem. (Sep_Empty)
∀P : setprop, {xEmpty|P x} = Empty
Proof:
Let P be given.
Apply Empty_eq to the current goal.
Let x be given.
Assume Hx.
An exact proof term for the current goal is EmptyE x (SepE1 Empty P x Hx).
Theorem. (Sep_Subq)
∀X : set, ∀P : setprop, {xX|P x} X
Proof:
An exact proof term for the current goal is SepE1.
Theorem. (Sep_In_Power)
∀X : set, ∀P : setprop, {xX|P x} 𝒫 X
Proof:
An exact proof term for the current goal is (λX P ⇒ PowerI X (Sep X P) (Sep_Subq X P)).
Definition. We define ReplSep to be λX P F ⇒ {F x|x{zX|P z}} of type set(setprop)(setset)set.
Notation. {B| xA, C} is notation for ReplSep Ax . C) (λ x . B).
Theorem. (ReplSepI)
∀X : set, ∀P : setprop, ∀F : setset, ∀x : set, x XP xF x {F x|xX, P x}
Proof:
An exact proof term for the current goal is (λX P F x u v ⇒ ReplI (Sep X P) F x (SepI X P x u v)).
Theorem. (ReplSepE)
∀X : set, ∀P : setprop, ∀F : setset, ∀y : set, y {F x|xX, P x}∃x : set, x X P x y = F x
Proof:
Let X, P, F and y be given.
Assume H1: y {F x|x{zX|P z}}.
Apply (ReplE {zX|P z} F y H1) to the current goal.
Let x be given.
Assume H2: x {zX|P z} y = F x.
Apply H2 to the current goal.
Assume H3: x {zX|P z}.
Assume H4: y = F x.
Apply (SepE X P x H3) to the current goal.
Assume H5: x X.
Assume H6: P x.
We use x to witness the existential quantifier.
Apply and3I to the current goal.
An exact proof term for the current goal is H5.
An exact proof term for the current goal is H6.
An exact proof term for the current goal is H4.
Theorem. (ReplSepE_impred)
∀X : set, ∀P : setprop, ∀F : setset, ∀y : set, y {F x|xX, P x}∀p : prop, (∀xX, P xy = F xp)p
Proof:
Let X, P, F and y be given.
Assume H1: y {F x|xX, P x}.
Let p be given.
Assume H2: ∀xX, P xy = F xp.
We will prove p.
Apply ReplSepE X P F y H1 to the current goal.
Let x be given.
Assume H3.
Apply H3 to the current goal.
Assume H3.
Apply H3 to the current goal.
An exact proof term for the current goal is H2 x.
Definition. We define binintersect to be λX Y ⇒ {xX|x Y} of type setsetset.
Notation. We use as an infix operator with priority 340 and which associates to the left corresponding to applying term binintersect.
Theorem. (binintersectI)
∀X Y z, z Xz Yz X Y
Proof:
An exact proof term for the current goal is (λX Y z H1 H2 ⇒ SepI X (λx : setx Y) z H1 H2).
Theorem. (binintersectE)
∀X Y z, z X Yz X z Y
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE X (λx : setx Y) z H1).
Theorem. (binintersectE1)
∀X Y z, z X Yz X
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE1 X (λx : setx Y) z H1).
Theorem. (binintersectE2)
∀X Y z, z X Yz Y
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE2 X (λx : setx Y) z H1).
Theorem. (binintersect_Subq_1)
∀X Y : set, X Y X
Proof:
An exact proof term for the current goal is binintersectE1.
Theorem. (binintersect_Subq_2)
∀X Y : set, X Y Y
Proof:
An exact proof term for the current goal is binintersectE2.
Theorem. (binintersect_Subq_eq_1)
∀X Y, X YX Y = X
Proof:
Let X and Y be given.
Assume H1: X Y.
Apply set_ext to the current goal.
Apply binintersect_Subq_1 to the current goal.
Let x be given.
Assume H2: x X.
Apply binintersectI to the current goal.
An exact proof term for the current goal is H2.
Apply H1 to the current goal.
An exact proof term for the current goal is H2.
Theorem. (binintersect_Subq_max)
∀X Y Z : set, Z XZ YZ X Y
Proof:
Let X, Y and Z be given.
Assume H1: Z X.
Assume H2: Z Y.
Let w be given.
Assume H3: w Z.
Apply (binintersectI X Y w) to the current goal.
We will prove w X.
An exact proof term for the current goal is (H1 w H3).
We will prove w Y.
An exact proof term for the current goal is (H2 w H3).
Theorem. (binintersect_com_Subq)
∀X Y : set, X Y Y X
Proof:
Let X and Y be given.
Apply (binintersect_Subq_max Y X (X Y)) to the current goal.
We will prove X Y Y.
Apply binintersect_Subq_2 to the current goal.
We will prove X Y X.
Apply binintersect_Subq_1 to the current goal.
Theorem. (binintersect_com)
∀X Y : set, X Y = Y X
Proof:
Let X and Y be given.
Apply set_ext to the current goal.
An exact proof term for the current goal is (binintersect_com_Subq X Y).
An exact proof term for the current goal is (binintersect_com_Subq Y X).
Definition. We define setminus to be λX Y ⇒ Sep X (λx ⇒ x Y) of type setsetset.
Notation. We use as an infix operator with priority 350 and no associativity corresponding to applying term setminus.
Theorem. (setminusI)
∀X Y z, (z X)(z Y)z X Y
Proof:
An exact proof term for the current goal is (λX Y z H1 H2 ⇒ SepI X (λx : setx Y) z H1 H2).
Theorem. (setminusE)
∀X Y z, (z X Y)z X z Y
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE X (λx : setx Y) z H).
Theorem. (setminusE1)
∀X Y z, (z X Y)z X
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE1 X (λx : setx Y) z H).
Theorem. (setminusE2)
∀X Y z, (z X Y)z Y
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE2 X (λx : setx Y) z H).
Theorem. (setminus_Subq)
∀X Y : set, X Y X
Proof:
An exact proof term for the current goal is setminusE1.
Theorem. (setminus_In_Power)
∀A U, A U 𝒫 A
Proof:
Let A and U be given.
Apply PowerI to the current goal.
Apply setminus_Subq to the current goal.
Theorem. (binunion_remove1_eq)
∀X, ∀xX, X = (X {x}) {x}
Proof:
Let X and x be given.
Assume Hx: x X.
Apply set_ext to the current goal.
Let y be given.
Assume Hy: y X.
We will prove y (X {x}) {x}.
Apply xm (y {x}) to the current goal.
Assume H1: y {x}.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H1.
Assume H1: y {x}.
Apply binunionI1 to the current goal.
Apply setminusI to the current goal.
An exact proof term for the current goal is Hy.
An exact proof term for the current goal is H1.
Let y be given.
Assume Hy: y (X {x}) {x}.
Apply binunionE (X {x}) {x} y Hy to the current goal.
Assume H1: y X {x}.
We will prove y X.
An exact proof term for the current goal is setminusE1 X {x} y H1.
Assume H1: y {x}.
We will prove y X.
rewrite the current goal using SingE x y H1 (from left to right).
We will prove x X.
An exact proof term for the current goal is Hx.
Theorem. (In_irref)
∀x, x x
Proof:
Apply In_ind to the current goal.
We will prove (∀X : set, (∀x : set, x Xx x)X X).
Let X be given.
Assume IH: ∀x : set, x Xx x.
Assume H: X X.
An exact proof term for the current goal is IH X H H.
Theorem. (In_no2cycle)
∀x y, x yy xFalse
Proof:
Apply In_ind to the current goal.
Let x be given.
Assume IH: ∀z, z x∀y, z yy zFalse.
Let y be given.
Assume H1: x y.
Assume H2: y x.
An exact proof term for the current goal is IH y H2 x H2 H1.
Definition. We define ordsucc to be λx : setx {x} of type setset.
Theorem. (ordsuccI1)
∀x : set, x ordsucc x
Proof:
Let x be given.
An exact proof term for the current goal is (λ(y : set)(H1 : y x) ⇒ binunionI1 x {x} y H1).
Theorem. (ordsuccI2)
∀x : set, x ordsucc x
Proof:
An exact proof term for the current goal is (λx : setbinunionI2 x {x} x (SingI x)).
Theorem. (ordsuccE)
∀x y : set, y ordsucc xy x y = x
Proof:
Let x and y be given.
Assume H1: y x {x}.
Apply (binunionE x {x} y H1) to the current goal.
Assume H2: y x.
Apply orIL to the current goal.
An exact proof term for the current goal is H2.
Assume H2: y {x}.
Apply orIR to the current goal.
An exact proof term for the current goal is (SingE x y H2).
Notation. Natural numbers 0,1,2,... are notation for the terms formed using Empty as 0 and forming successors with ordsucc.
Theorem. (neq_0_ordsucc)
∀a : set, 0 ordsucc a
Proof:
Let a be given.
We will prove ¬ (0 = ordsucc a).
Assume H1: 0 = ordsucc a.
We prove the intermediate claim L1: a ordsucc aFalse.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is (EmptyE a).
An exact proof term for the current goal is (L1 (ordsuccI2 a)).
Theorem. (neq_ordsucc_0)
∀a : set, ordsucc a 0
Proof:
Let a be given.
An exact proof term for the current goal is neq_i_sym 0 (ordsucc a) (neq_0_ordsucc a).
Theorem. (ordsucc_inj)
∀a b : set, ordsucc a = ordsucc ba = b
Proof:
Let a and b be given.
Assume H1: ordsucc a = ordsucc b.
We prove the intermediate claim L1: a ordsucc b.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is (ordsuccI2 a).
Apply (ordsuccE b a L1) to the current goal.
Assume H2: a b.
We prove the intermediate claim L2: b ordsucc a.
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is (ordsuccI2 b).
Apply (ordsuccE a b L2) to the current goal.
Assume H3: b a.
We will prove False.
An exact proof term for the current goal is (In_no2cycle a b H2 H3).
Assume H3: b = a.
Use symmetry.
An exact proof term for the current goal is H3.
Assume H2: a = b.
An exact proof term for the current goal is H2.
Theorem. (In_0_1)
Proof:
An exact proof term for the current goal is (ordsuccI2 0).
Theorem. (In_0_2)
Proof:
An exact proof term for the current goal is (ordsuccI1 1 0 In_0_1).
Theorem. (In_1_2)
Proof:
An exact proof term for the current goal is (ordsuccI2 1).
Definition. We define nat_p to be λn : set∀p : setprop, p 0(∀x : set, p xp (ordsucc x))p n of type setprop.
Theorem. (nat_0)
Proof:
An exact proof term for the current goal is (λp H _ ⇒ H).
Theorem. (nat_ordsucc)
∀n : set, nat_p nnat_p (ordsucc n)
Proof:
An exact proof term for the current goal is (λn H1 p H2 H3 ⇒ H3 n (H1 p H2 H3)).
Theorem. (nat_1)
Proof:
An exact proof term for the current goal is (nat_ordsucc 0 nat_0).
Theorem. (nat_2)
Proof:
An exact proof term for the current goal is (nat_ordsucc 1 nat_1).
Theorem. (nat_0_in_ordsucc)
∀n, nat_p n0 ordsucc n
Proof:
Let n be given.
Assume H1.
Apply H1 (λn ⇒ 0 ordsucc n) to the current goal.
We will prove 0 ordsucc 0.
An exact proof term for the current goal is In_0_1.
Let n be given.
Assume IH: 0 ordsucc n.
We will prove 0 ordsucc (ordsucc n).
An exact proof term for the current goal is (ordsuccI1 (ordsucc n) 0 IH).
Theorem. (nat_ordsucc_in_ordsucc)
∀n, nat_p n∀mn, ordsucc m ordsucc n
Proof:
Let n be given.
Assume H1.
Apply (H1 (λn ⇒ ∀mn, ordsucc m ordsucc n)) to the current goal.
We will prove ∀m0, ordsucc m ordsucc 0.
Let m be given.
Assume Hm: m 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume IH: ∀mn, ordsucc m ordsucc n.
We will prove ∀mordsucc n, ordsucc m ordsucc (ordsucc n).
Let m be given.
Assume H2: m ordsucc n.
We will prove ordsucc m ordsucc (ordsucc n).
Apply (ordsuccE n m H2) to the current goal.
Assume H3: m n.
We prove the intermediate claim L1: ordsucc m ordsucc n.
An exact proof term for the current goal is (IH m H3).
An exact proof term for the current goal is (ordsuccI1 (ordsucc n) (ordsucc m) L1).
Assume H3: m = n.
rewrite the current goal using H3 (from left to right).
We will prove ordsucc n ordsucc (ordsucc n).
An exact proof term for the current goal is (ordsuccI2 (ordsucc n)).
Theorem. (nat_ind)
∀p : setprop, p 0(∀n, nat_p np np (ordsucc n))∀n, nat_p np n
Proof:
Let p be given.
Assume H1: p 0.
Assume H2: ∀n, nat_p np np (ordsucc n).
We prove the intermediate claim L1: nat_p 0 p 0.
An exact proof term for the current goal is (andI (nat_p 0) (p 0) nat_0 H1).
We prove the intermediate claim L2: ∀n, nat_p n p nnat_p (ordsucc n) p (ordsucc n).
Let n be given.
Assume H3: nat_p n p n.
Apply H3 to the current goal.
Assume H4: nat_p n.
Assume H5: p n.
Apply andI to the current goal.
We will prove nat_p (ordsucc n).
An exact proof term for the current goal is (nat_ordsucc n H4).
We will prove p (ordsucc n).
An exact proof term for the current goal is (H2 n H4 H5).
Let n be given.
Assume H3.
We prove the intermediate claim L3: nat_p n p n.
An exact proof term for the current goal is (H3 (λn ⇒ nat_p n p n) L1 L2).
An exact proof term for the current goal is (andER (nat_p n) (p n) L3).
Theorem. (nat_complete_ind)
∀p : setprop, (∀n, nat_p n(∀mn, p m)p n)∀n, nat_p np n
Proof:
Let p be given.
Assume H1: ∀n, nat_p n(∀mn, p m)p n.
We prove the intermediate claim L1: ∀n : set, nat_p n∀mn, p m.
Apply nat_ind to the current goal.
We will prove ∀m0, p m.
Let m be given.
Assume Hm: m 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀mn, p m.
We will prove ∀mordsucc n, p m.
Let m be given.
Assume Hm: m ordsucc n.
We will prove p m.
Apply (ordsuccE n m Hm) to the current goal.
Assume H2: m n.
An exact proof term for the current goal is (IHn m H2).
Assume H2: m = n.
We will prove p m.
rewrite the current goal using H2 (from left to right).
We will prove p n.
An exact proof term for the current goal is (H1 n Hn IHn).
We will prove ∀n, nat_p np n.
An exact proof term for the current goal is (λn Hn ⇒ H1 n Hn (L1 n Hn)).
Theorem. (nat_inv_impred)
∀p : setprop, p 0(∀n, nat_p np (ordsucc n))∀n, nat_p np n
Proof:
Let p be given.
Assume H1 H2.
An exact proof term for the current goal is nat_ind p H1 (λn H _ ⇒ H2 n H).
Theorem. (nat_inv)
∀n, nat_p nn = 0 ∃x, nat_p x n = ordsucc x
Proof:
Apply nat_inv_impred to the current goal.
Apply orIL to the current goal.
Use reflexivity.
Let n be given.
Assume Hn.
Apply orIR to the current goal.
We use n to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hn.
Use reflexivity.
Theorem. (nat_p_trans)
∀n, nat_p n∀mn, nat_p m
Proof:
Apply nat_ind to the current goal.
We will prove ∀m0, nat_p m.
Let m be given.
Assume Hm: m 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀mn, nat_p m.
We will prove ∀mordsucc n, nat_p m.
Let m be given.
Assume Hm: m ordsucc n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m n.
An exact proof term for the current goal is (IHn m H1).
Assume H1: m = n.
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is Hn.
Theorem. (nat_trans)
∀n, nat_p n∀mn, m n
Proof:
Apply nat_ind to the current goal.
We will prove ∀m0, m 0.
Let m be given.
Assume Hm: m 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀mn, m n.
We will prove ∀mordsucc n, m ordsucc n.
Let m be given.
Assume Hm: m ordsucc n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m n.
We will prove m ordsucc n.
Apply (Subq_tra m n (ordsucc n)) to the current goal.
An exact proof term for the current goal is (IHn m H1).
An exact proof term for the current goal is (ordsuccI1 n).
Assume H1: m = n.
We will prove m ordsucc n.
rewrite the current goal using H1 (from left to right).
We will prove n ordsucc n.
An exact proof term for the current goal is (ordsuccI1 n).
Theorem. (nat_ordsucc_trans)
∀n, nat_p n∀mordsucc n, m n
Proof:
Let n be given.
Assume Hn: nat_p n.
Let m be given.
Assume Hm: m ordsucc n.
Let k be given.
Assume Hk: k m.
We will prove k n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m n.
An exact proof term for the current goal is nat_trans n Hn m H1 k Hk.
Assume H1: m = n.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is Hk.
Definition. We define surj to be λX Y f ⇒ (∀uX, f u Y) (∀wY, ∃uX, f u = w) of type setset(setset)prop.
Theorem. (form100_63_surjCantor)
∀A : set, ∀f : setset, ¬ surj A (𝒫 A) f
Proof:
Let A and f be given.
Assume H.
Apply H to the current goal.
Assume H1: ∀uA, f u 𝒫 A.
Assume H2: ∀w𝒫 A, ∃uA, f u = w.
Set D to be the term {xA|x f x}.
We prove the intermediate claim L1: D 𝒫 A.
An exact proof term for the current goal is Sep_In_Power A (λx ⇒ x f x).
Apply H2 D L1 to the current goal.
Let d be given.
Assume H.
Apply H to the current goal.
Assume Hd: d A.
Assume HfdD: f d = D.
We prove the intermediate claim L2: d D.
Assume H3: d D.
Apply SepE2 A (λx ⇒ x f x) d H3 to the current goal.
We will prove d f d.
rewrite the current goal using HfdD (from left to right).
We will prove d D.
An exact proof term for the current goal is H3.
Apply L2 to the current goal.
We will prove d D.
Apply SepI to the current goal.
We will prove d A.
An exact proof term for the current goal is Hd.
We will prove d f d.
rewrite the current goal using HfdD (from left to right).
An exact proof term for the current goal is L2.
Definition. We define inj to be λX Y f ⇒ (∀uX, f u Y) (∀u vX, f u = f vu = v) of type setset(setset)prop.
Theorem. (form100_63_injCantor)
∀A : set, ∀f : setset, ¬ inj (𝒫 A) A f
Proof:
Let A and f be given.
Assume H.
Apply H to the current goal.
Assume H1: ∀X𝒫 A, f X A.
Assume H2: ∀X Y𝒫 A, f X = f YX = Y.
Set D to be the term {f X|X𝒫 A, f X X}.
We prove the intermediate claim L1: D 𝒫 A.
Apply PowerI to the current goal.
Let n be given.
Assume H3: n D.
Apply ReplSepE_impred (𝒫 A) (λX ⇒ f X X) f n H3 to the current goal.
Let X be given.
Assume HX: X 𝒫 A.
Assume H4: f X X.
Assume H5: n = f X.
We will prove n A.
rewrite the current goal using H5 (from left to right).
Apply H1 to the current goal.
An exact proof term for the current goal is HX.
We prove the intermediate claim L2: f D D.
Assume H3: f D D.
Apply ReplSepE_impred (𝒫 A) (λX ⇒ f X X) f (f D) H3 to the current goal.
Let X be given.
Assume HX: X 𝒫 A.
Assume H4: f X X.
Assume H5: f D = f X.
We prove the intermediate claim L2a: D = X.
An exact proof term for the current goal is H2 D L1 X HX H5.
Apply H4 to the current goal.
rewrite the current goal using L2a (from right to left).
An exact proof term for the current goal is H3.
Apply L2 to the current goal.
We will prove f D D.
Apply ReplSepI to the current goal.
We will prove D 𝒫 A.
An exact proof term for the current goal is L1.
We will prove f D D.
An exact proof term for the current goal is L2.
Theorem. (injI)
∀X Y, ∀f : setset, (∀xX, f x Y)(∀x zX, f x = f zx = z)inj X Y f
Proof:
Let X, Y and f be given.
Assume H1 H2.
We will prove (∀xX, f x Y) (∀x zX, f x = f zx = z).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Theorem. (inj_comp)
∀X Y Z : set, ∀f g : setset, inj X Y finj Y Z ginj X Z (λx ⇒ g (f x))
Proof:
Let X, Y, Z, f and g be given.
Assume Hf.
Assume Hg.
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
Apply injI to the current goal.
Let u be given.
Assume Hu: u X.
An exact proof term for the current goal is (Hg1 (f u) (Hf1 u Hu)).
Let u be given.
Assume Hu: u X.
Let v be given.
Assume Hv: v X.
Assume H1: g (f u) = g (f v).
We will prove u = v.
Apply Hf2 u Hu v Hv to the current goal.
We will prove f u = f v.
Apply Hg2 (f u) (Hf1 u Hu) (f v) (Hf1 v Hv) to the current goal.
We will prove g (f u) = g (f v).
An exact proof term for the current goal is H1.
Definition. We define bij to be λX Y f ⇒ (∀uX, f u Y) (∀u vX, f u = f vu = v) (∀wY, ∃uX, f u = w) of type setset(setset)prop.
Theorem. (bijI)
∀X Y, ∀f : setset, (∀uX, f u Y)(∀u vX, f u = f vu = v)(∀wY, ∃uX, f u = w)bij X Y f
Proof:
Let X, Y and f be given.
Assume Hf1 Hf2 Hf3.
We will prove (∀uX, f u Y) (∀u vX, f u = f vu = v) (∀wY, ∃uX, f u = w).
Apply and3I to the current goal.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is Hf2.
An exact proof term for the current goal is Hf3.
Theorem. (bijE)
∀X Y, ∀f : setset, bij X Y f∀p : prop, ((∀uX, f u Y)(∀u vX, f u = f vu = v)(∀wY, ∃uX, f u = w)p)p
Proof:
Let X, Y and f be given.
Assume Hf.
Let p be given.
Assume Hp.
Apply Hf to the current goal.
Assume Hf.
Apply Hf to the current goal.
Assume Hf1 Hf2 Hf3.
An exact proof term for the current goal is Hp Hf1 Hf2 Hf3.
Theorem. (bij_inj)
∀X Y, ∀f : setset, bij X Y finj X Y f
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H1 _.
An exact proof term for the current goal is H1.
Theorem. (bij_id)
∀X, bij X X (λx ⇒ x)
Proof:
Let X be given.
We will prove (∀uX, u X) (∀u vX, u = vu = v) (∀wX, ∃uX, u = w).
Apply and3I to the current goal.
An exact proof term for the current goal is (λu Hu ⇒ Hu).
An exact proof term for the current goal is (λu Hu v Hv H1 ⇒ H1).
Let w be given.
Assume Hw.
We use w to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hw.
Use reflexivity.
Theorem. (bij_comp)
∀X Y Z : set, ∀f g : setset, bij X Y fbij Y Z gbij X Z (λx ⇒ g (f x))
Proof:
Let X, Y, Z, f and g be given.
Assume Hf.
Apply Hf to the current goal.
Assume Hf12 Hf3.
Apply Hf12 to the current goal.
Assume Hf1 Hf2.
Assume Hg.
Apply Hg to the current goal.
Assume Hg12 Hg3.
Apply Hg12 to the current goal.
Assume Hg1 Hg2.
We will prove (∀uX, g (f u) Z) (∀u vX, g (f u) = g (f v)u = v) (∀wZ, ∃uX, g (f u) = w).
Apply and3I to the current goal.
Let u be given.
Assume Hu: u X.
An exact proof term for the current goal is (Hg1 (f u) (Hf1 u Hu)).
Let u be given.
Assume Hu: u X.
Let v be given.
Assume Hv: v X.
Assume H1: g (f u) = g (f v).
We will prove u = v.
Apply Hf2 u Hu v Hv to the current goal.
We will prove f u = f v.
Apply Hg2 (f u) (Hf1 u Hu) (f v) (Hf1 v Hv) to the current goal.
We will prove g (f u) = g (f v).
An exact proof term for the current goal is H1.
Let w be given.
Assume Hw: w Z.
Apply Hg3 w Hw to the current goal.
Let y be given.
Assume Hy12.
Apply Hy12 to the current goal.
Assume Hy1: y Y.
Assume Hy2: g y = w.
Apply Hf3 y Hy1 to the current goal.
Let u be given.
Assume Hu12.
Apply Hu12 to the current goal.
Assume Hu1: u X.
Assume Hu2: f u = y.
We will prove ∃uX, g (f u) = w.
We use u to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hu1.
rewrite the current goal using Hu2 (from left to right).
An exact proof term for the current goal is Hy2.
Theorem. (bij_surj)
∀X Y, ∀f : setset, bij X Y fsurj X Y f
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1 _ H2.
We will prove (∀uX, f u Y) (∀wY, ∃uX, f u = w).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Definition. We define inv to be λX f ⇒ λy : setEps_i (λx ⇒ x X f x = y) of type set(setset)setset.
Theorem. (surj_rinv)
∀X Y, ∀f : setset, (∀wY, ∃uX, f u = w)∀yY, inv X f y X f (inv X f y) = y
Proof:
Let X, Y and f be given.
Assume H1.
Let y be given.
Assume Hy: y Y.
Apply H1 y Hy to the current goal.
Let x be given.
Assume H2.
An exact proof term for the current goal is Eps_i_ax (λx ⇒ x X f x = y) x H2.
Theorem. (inj_linv)
∀X, ∀f : setset, (∀u vX, f u = f vu = v)∀xX, inv X f (f x) = x
Proof:
Let X and f be given.
Assume H1.
Let x be given.
Assume Hx.
We prove the intermediate claim L1: inv X f (f x) X f (inv X f (f x)) = f x.
Apply Eps_i_ax (λx' ⇒ x' X f x' = f x) x to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is Hx.
Use reflexivity.
Apply L1 to the current goal.
Assume H2 H3.
An exact proof term for the current goal is H1 (inv X f (f x)) H2 x Hx H3.
Theorem. (bij_inv)
∀X Y, ∀f : setset, bij X Y fbij Y X (inv X f)
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H2.
Apply H2 to the current goal.
Assume H3: ∀uX, f u Y.
Assume H4: ∀u vX, f u = f vu = v.
Assume H5: ∀wY, ∃uX, f u = w.
Set g to be the term λy ⇒ Eps_i (λx ⇒ x X f x = y) of type setset.
We prove the intermediate claim L1: ∀yY, g y X f (g y) = y.
An exact proof term for the current goal is surj_rinv X Y f H5.
We will prove (∀uY, g u X) (∀u vY, g u = g vu = v) (∀wX, ∃uY, g u = w).
Apply and3I to the current goal.
We will prove ∀uY, g u X.
Let u be given.
Assume Hu.
Apply L1 u Hu to the current goal.
Assume H _.
An exact proof term for the current goal is H.
We will prove ∀u vY, g u = g vu = v.
Let u be given.
Assume Hu.
Let v be given.
Assume Hv H6.
We will prove u = v.
Apply L1 u Hu to the current goal.
Assume H7: g u X.
Assume H8: f (g u) = u.
Apply L1 v Hv to the current goal.
Assume H9: g v X.
Assume H10: f (g v) = v.
rewrite the current goal using H8 (from right to left).
rewrite the current goal using H10 (from right to left).
rewrite the current goal using H6 (from right to left).
Use reflexivity.
We will prove ∀wX, ∃uY, g u = w.
Let w be given.
Assume Hw.
We prove the intermediate claim Lfw: f w Y.
An exact proof term for the current goal is H3 w Hw.
We use f w to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Lfw.
An exact proof term for the current goal is inj_linv X f H4 w Hw.
Definition. We define atleastp to be λX Y : set∃f : setset, inj X Y f of type setsetprop.
Theorem. (atleastp_tra)
∀X Y Z, atleastp X Yatleastp Y Zatleastp X Z
Proof:
The rest of this subproof is missing.
Theorem. (Subq_atleastp)
∀X Y, X Yatleastp X Y
Proof:
The rest of this subproof is missing.
Definition. We define equip to be λX Y : set∃f : setset, bij X Y f of type setsetprop.
Theorem. (equip_atleastp)
∀X Y, equip X Yatleastp X Y
Proof:
The rest of this subproof is missing.
Theorem. (equip_ref)
∀X, equip X X
Proof:
The rest of this subproof is missing.
Theorem. (equip_sym)
∀X Y, equip X Yequip Y X
Proof:
The rest of this subproof is missing.
Theorem. (equip_tra)
∀X Y Z, equip X Yequip Y Zequip X Z
Proof:
The rest of this subproof is missing.
Theorem. (equip_0_Empty)
∀X, equip X 0X = 0
Proof:
The rest of this subproof is missing.
Theorem. (equip_adjoin_ordsucc)
∀N X y, y Xequip N Xequip (ordsucc N) (X {y})
Proof:
The rest of this subproof is missing.
Theorem. (equip_ordsucc_remove1)
∀X N, ∀xX, equip X (ordsucc N)equip (X {x}) N
Proof:
The rest of this subproof is missing.
Beginning of Section SchroederBernstein
Theorem. (KnasterTarski_set)
∀A, ∀F : setset, (∀U𝒫 A, F U 𝒫 A)(∀U V𝒫 A, U VF U F V)∃Y𝒫 A, F Y = Y
Proof:
The rest of this subproof is missing.
Theorem. (image_In_Power)
∀A B, ∀f : setset, (∀xA, f x B)∀U𝒫 A, {f x|xU} 𝒫 B
Proof:
The rest of this subproof is missing.
Theorem. (image_monotone)
∀f : setset, ∀U V, U V{f x|xU} {f x|xV}
Proof:
The rest of this subproof is missing.
Theorem. (setminus_antimonotone)
∀A U V, U VA V A U
Proof:
The rest of this subproof is missing.
Theorem. (SchroederBernstein)
∀A B, ∀f g : setset, inj A B finj B A gequip A B
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SchroederBernstein
Beginning of Section PigeonHole
Theorem. (PigeonHole_nat)
∀n, nat_p n∀f : setset, (∀iordsucc n, f i n)¬ (∀i jordsucc n, f i = f ji = j)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section PigeonHole
Theorem. (Union_ordsucc_eq)
∀n, nat_p n (ordsucc n) = n
Proof:
The rest of this subproof is missing.
Theorem. (cases_1)
∀i1, ∀p : setprop, p 0p i
Proof:
The rest of this subproof is missing.
Theorem. (cases_2)
∀i2, ∀p : setprop, p 0p 1p i
Proof:
The rest of this subproof is missing.
Theorem. (neq_0_1)
Proof:
The rest of this subproof is missing.
Theorem. (neq_1_0)
Proof:
The rest of this subproof is missing.
Theorem. (neq_0_2)
Proof:
The rest of this subproof is missing.
Theorem. (neq_2_0)
Proof:
The rest of this subproof is missing.
Definition. We define ordinal to be λalpha : setTransSet alpha ∀betaalpha, TransSet beta of type setprop.
Theorem. (ordinal_TransSet)
∀alpha : set, ordinal alphaTransSet alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_Empty)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_Hered)
∀alpha : set, ordinal alpha∀betaalpha, ordinal beta
Proof:
The rest of this subproof is missing.
Theorem. (TransSet_ordsucc)
∀X : set, TransSet XTransSet (ordsucc X)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc)
∀alpha : set, ordinal alphaordinal (ordsucc alpha)
Proof:
The rest of this subproof is missing.
Theorem. (nat_p_ordinal)
∀n : set, nat_p nordinal n
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_1)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_2)
Proof:
The rest of this subproof is missing.
Theorem. (TransSet_ordsucc_In_Subq)
∀X : set, TransSet X∀xX, ordsucc x X
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_In_Subq)
∀alpha, ordinal alpha∀betaalpha, ordsucc beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_trichotomy_or)
∀alpha beta : set, ordinal alphaordinal betaalpha beta alpha = beta beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_trichotomy_or_impred)
∀alpha beta : set, ordinal alphaordinal beta∀p : prop, (alpha betap)(alpha = betap)(beta alphap)p
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_In_Or_Subq)
∀alpha beta, ordinal alphaordinal betaalpha beta beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_linear)
∀alpha beta, ordinal alphaordinal betaalpha beta beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_In_eq)
∀alpha beta, ordinal alphabeta alphaordsucc beta alpha alpha = ordsucc beta
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_lim_or_succ)
∀alpha, ordinal alpha(∀betaalpha, ordsucc beta alpha) (∃betaalpha, alpha = ordsucc beta)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_In)
∀alpha, ordinal alpha∀betaalpha, ordsucc beta ordsucc alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_famunion)
∀X, ∀F : setset, (∀xX, ordinal (F x))ordinal (xXF x)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_binintersect)
∀alpha beta, ordinal alphaordinal betaordinal (alpha beta)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_binunion)
∀alpha beta, ordinal alphaordinal betaordinal (alpha beta)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ind)
∀p : setprop, (∀alpha, ordinal alpha(∀betaalpha, p beta)p alpha)∀alpha, ordinal alphap alpha
Proof:
The rest of this subproof is missing.
Theorem. (least_ordinal_ex)
∀p : setprop, (∃alpha, ordinal alpha p alpha)∃alpha, ordinal alpha p alpha ∀betaalpha, ¬ p beta
Proof:
The rest of this subproof is missing.
Theorem. (equip_Sing_1)
∀x, equip {x} 1
Proof:
The rest of this subproof is missing.
Theorem. (TransSet_In_ordsucc_Subq)
∀x y, TransSet yx ordsucc yx y
Proof:
The rest of this subproof is missing.
Theorem. (exandE_i)
∀P Q : setprop, (∃x, P x Q x)∀r : prop, (∀x, P xQ xr)r
Proof:
The rest of this subproof is missing.
Theorem. (exandE_ii)
∀P Q : (setset)prop, (∃x : setset, P x Q x)∀p : prop, (∀x : setset, P xQ xp)p
Proof:
The rest of this subproof is missing.
Theorem. (exandE_iii)
∀P Q : (setsetset)prop, (∃x : setsetset, P x Q x)∀p : prop, (∀x : setsetset, P xQ xp)p
Proof:
The rest of this subproof is missing.
Theorem. (exandE_iiii)
∀P Q : (setsetsetset)prop, (∃x : setsetsetset, P x Q x)∀p : prop, (∀x : setsetsetset, P xQ xp)p
Proof:
The rest of this subproof is missing.
Beginning of Section Descr_ii
Variable P : (setset)prop
Definition. We define Descr_ii to be λx : setEps_i (λy ⇒ ∀h : setset, P hh x = y) of type setset.
Hypothesis Pex : ∃f : setset, P f
Hypothesis Puniq : ∀f g : setset, P fP gf = g
Proof:
The rest of this subproof is missing.
End of Section Descr_ii
Beginning of Section Descr_iii
Variable P : (setsetset)prop
Definition. We define Descr_iii to be λx y : setEps_i (λz ⇒ ∀h : setsetset, P hh x y = z) of type setsetset.
Hypothesis Pex : ∃f : setsetset, P f
Hypothesis Puniq : ∀f g : setsetset, P fP gf = g
Proof:
The rest of this subproof is missing.
End of Section Descr_iii
Beginning of Section Descr_Vo1
Variable P : Vo 1prop
Definition. We define Descr_Vo1 to be λx : set∀h : setprop, P hh x of type Vo 1.
Hypothesis Pex : ∃f : Vo 1, P f
Hypothesis Puniq : ∀f g : Vo 1, P fP gf = g
Proof:
The rest of this subproof is missing.
End of Section Descr_Vo1
Beginning of Section If_ii
Variable p : prop
Variable f g : setset
Definition. We define If_ii to be λx ⇒ if p then f x else g x of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section If_ii
Beginning of Section If_iii
Variable p : prop
Variable f g : setsetset
Definition. We define If_iii to be λx y ⇒ if p then f x y else g x y of type setsetset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section If_iii
Beginning of Section EpsilonRec_i
Variable F : set(setset)set
Definition. We define In_rec_i_G to be λX Y ⇒ ∀R : setsetprop, (∀X : set, ∀f : setset, (∀xX, R x (f x))R X (F X f))R X Y of type setsetprop.
Definition. We define In_rec_i to be λX ⇒ Eps_i (In_rec_i_G X) of type setset.
Theorem. (In_rec_i_G_c)
∀X : set, ∀f : setset, (∀xX, In_rec_i_G x (f x))In_rec_i_G X (F X f)
Proof:
The rest of this subproof is missing.
Theorem. (In_rec_i_G_inv)
∀X : set, ∀Y : set, In_rec_i_G X Y∃f : setset, (∀xX, In_rec_i_G x (f x)) Y = F X f
Proof:
The rest of this subproof is missing.
Hypothesis Fr : ∀X : set, ∀g h : setset, (∀xX, g x = h x)F X g = F X h
Theorem. (In_rec_i_G_f)
∀X : set, ∀Y Z : set, In_rec_i_G X YIn_rec_i_G X ZY = Z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
Variable F : set(set(setset))(setset)
Definition. We define In_rec_G_ii to be λX Y ⇒ ∀R : set(setset)prop, (∀X : set, ∀f : set(setset), (∀xX, R x (f x))R X (F X f))R X Y of type set(setset)prop.
Definition. We define In_rec_ii to be λX ⇒ Descr_ii (In_rec_G_ii X) of type set(setset).
Theorem. (In_rec_G_ii_c)
∀X : set, ∀f : set(setset), (∀xX, In_rec_G_ii x (f x))In_rec_G_ii X (F X f)
Proof:
The rest of this subproof is missing.
Theorem. (In_rec_G_ii_inv)
∀X : set, ∀Y : (setset), In_rec_G_ii X Y∃f : set(setset), (∀xX, In_rec_G_ii x (f x)) Y = F X f
Proof:
The rest of this subproof is missing.
Hypothesis Fr : ∀X : set, ∀g h : set(setset), (∀xX, g x = h x)F X g = F X h
Theorem. (In_rec_G_ii_f)
∀X : set, ∀Y Z : (setset), In_rec_G_ii X YIn_rec_G_ii X ZY = Z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
Variable F : set(set(setsetset))(setsetset)
Definition. We define In_rec_G_iii to be λX Y ⇒ ∀R : set(setsetset)prop, (∀X : set, ∀f : set(setsetset), (∀xX, R x (f x))R X (F X f))R X Y of type set(setsetset)prop.
Definition. We define In_rec_iii to be λX ⇒ Descr_iii (In_rec_G_iii X) of type set(setsetset).
Theorem. (In_rec_G_iii_c)
∀X : set, ∀f : set(setsetset), (∀xX, In_rec_G_iii x (f x))In_rec_G_iii X (F X f)
Proof:
The rest of this subproof is missing.
Theorem. (In_rec_G_iii_inv)
∀X : set, ∀Y : (setsetset), In_rec_G_iii X Y∃f : set(setsetset), (∀xX, In_rec_G_iii x (f x)) Y = F X f
Proof:
The rest of this subproof is missing.
Hypothesis Fr : ∀X : set, ∀g h : set(setsetset), (∀xX, g x = h x)F X g = F X h
Theorem. (In_rec_G_iii_f)
∀X : set, ∀Y Z : (setsetset), In_rec_G_iii X YIn_rec_G_iii X ZY = Z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section EpsilonRec_iii
Beginning of Section NatRec
Variable z : set
Variable f : setsetset
Let F : set(setset)setλn g ⇒ if n n then f ( n) (g ( n)) else z
Definition. We define nat_primrec to be In_rec_i F of type setset.
Theorem. (nat_primrec_r)
∀X : set, ∀g h : setset, (∀xX, g x = h x)F X g = F X h
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section NatRec
Beginning of Section NatAdd
Definition. We define add_nat to be λn m : setnat_primrec n (λ_ r ⇒ ordsucc r) m of type setsetset.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Theorem. (add_nat_0R)
∀n : set, n + 0 = n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_SR)
∀n m : set, nat_p mn + ordsucc m = ordsucc (n + m)
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_p)
∀n : set, nat_p n∀m : set, nat_p mnat_p (n + m)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_asso)
∀n : set, nat_p n∀m : set, nat_p m∀k : set, nat_p k(n + m) + k = n + (m + k)
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_0L)
∀m : set, nat_p m0 + m = m
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_SL)
∀n : set, nat_p n∀m : set, nat_p mordsucc n + m = ordsucc (n + m)
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_com)
∀n : set, nat_p n∀m : set, nat_p mn + m = m + n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_In_R)
∀m, nat_p m∀km, ∀n, nat_p nk + n m + n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_In_L)
∀n, nat_p n∀m, nat_p m∀km, n + k n + m
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_Subq_R)
∀k, nat_p k∀m, nat_p mk m∀n, nat_p nk + n m + n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_Subq_L)
∀n, nat_p n∀k, nat_p k∀m, nat_p mk mn + k n + m
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_Subq_R')
∀m, nat_p m∀n, nat_p nm m + n
Proof:
The rest of this subproof is missing.
Theorem. (nat_Subq_add_ex)
∀n, nat_p n∀m, nat_p mn m∃k, nat_p k m = k + n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_cancel_R)
∀k, nat_p k∀m, nat_p m∀n, nat_p nk + n = m + nk = m
Proof:
The rest of this subproof is missing.
End of Section NatAdd
Beginning of Section NatMul
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Definition. We define mul_nat to be λn m : setnat_primrec 0 (λ_ r ⇒ n + r) m of type setsetset.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (mul_nat_0R)
∀n : set, n * 0 = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_SR)
∀n m, nat_p mn * ordsucc m = n + n * m
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_p)
∀n : set, nat_p n∀m : set, nat_p mnat_p (n * m)
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_0L)
∀m : set, nat_p m0 * m = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_SL)
∀n : set, nat_p n∀m : set, nat_p mordsucc n * m = n * m + m
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_com)
∀n : set, nat_p n∀m : set, nat_p mn * m = m * n
Proof:
The rest of this subproof is missing.
Theorem. (mul_add_nat_distrL)
∀n : set, nat_p n∀m : set, nat_p m∀k : set, nat_p kn * (m + k) = n * m + n * k
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_asso)
∀n : set, nat_p n∀m : set, nat_p m∀k : set, nat_p k(n * m) * k = n * (m * k)
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_Subq_R)
∀m n, nat_p mnat_p nm n∀k, nat_p km * k n * k
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_Subq_L)
∀m n, nat_p mnat_p nm n∀k, nat_p kk * m k * n
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_0_or_Subq)
∀m, nat_p m∀n, nat_p nn = 0 m m * n
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_0_inv)
∀m, nat_p m∀n, nat_p nm * n = 0m = 0 n = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_0m_1n_In)
∀m, nat_p m∀n, nat_p n0 m1 nm m * n
Proof:
The rest of this subproof is missing.
Theorem. (nat_le1_cases)
∀m, nat_p mm 1m = 0 m = 1
Proof:
The rest of this subproof is missing.
Definition. We define Pi_nat to be λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type (setset)setset.
Theorem. (Pi_nat_0)
∀f : setset, Pi_nat f 0 = 1
Proof:
The rest of this subproof is missing.
Theorem. (Pi_nat_S)
∀f : setset, ∀n, nat_p nPi_nat f (ordsucc n) = Pi_nat f n * f n
Proof:
The rest of this subproof is missing.
Theorem. (Pi_nat_p)
∀f : setset, ∀n, nat_p n(∀in, nat_p (f i))nat_p (Pi_nat f n)
Proof:
The rest of this subproof is missing.
Theorem. (Pi_nat_0_inv)
∀f : setset, ∀n, nat_p n(∀in, nat_p (f i))Pi_nat f n = 0(∃in, f i = 0)
Proof:
The rest of this subproof is missing.
Definition. We define exp_nat to be λn m : setnat_primrec 1 (λ_ r ⇒ n * r) m of type setsetset.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.
Proof:
The rest of this subproof is missing.
Theorem. (exp_nat_S)
∀n m, nat_p mn ^ (ordsucc m) = n * n ^ m
Proof:
The rest of this subproof is missing.
Theorem. (exp_nat_p)
∀n, nat_p n∀m, nat_p mnat_p (n ^ m)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section NatMul
Beginning of Section form100_52
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (equip_finite_Power)
∀n, nat_p n∀X, equip X nequip (𝒫 X) (2 ^ n)
Proof:
The rest of this subproof is missing.
End of Section form100_52
Theorem. (ZF_closed_E)
∀U, ZF_closed U∀p : prop, (Union_closed UPower_closed URepl_closed Up)p
Proof:
The rest of this subproof is missing.
Theorem. (ZF_Union_closed)
∀U, ZF_closed U∀XU, X U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_Power_closed)
∀U, ZF_closed U∀XU, 𝒫 X U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_Repl_closed)
∀U, ZF_closed U∀XU, ∀F : setset, (∀xX, F x U){F x|xX} U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_UPair_closed)
∀U, ZF_closed U∀x yU, {x,y} U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_Sing_closed)
∀U, ZF_closed U∀xU, {x} U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_binunion_closed)
∀U, ZF_closed U∀X YU, (X Y) U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_ordsucc_closed)
∀U, ZF_closed U∀xU, ordsucc x U
Proof:
The rest of this subproof is missing.
Theorem. (nat_p_UnivOf_Empty)
∀n : set, nat_p nn UnivOf Empty
Proof:
The rest of this subproof is missing.
Definition. We define ω to be {nUnivOf Empty|nat_p n} of type set.
Theorem. (omega_nat_p)
∀nω, nat_p n
Proof:
The rest of this subproof is missing.
Theorem. (nat_p_omega)
∀n : set, nat_p nn ω
Proof:
The rest of this subproof is missing.
Theorem. (omega_ordsucc)
Proof:
The rest of this subproof is missing.
Theorem. (form100_22_v2)
∀f : setset, ¬ inj (𝒫 ω) ω f
Proof:
The rest of this subproof is missing.
Theorem. (form100_22_v3)
∀g : setset, ¬ surj ω (𝒫 ω) g
Proof:
The rest of this subproof is missing.
Theorem. (form100_22_v1)
Proof:
The rest of this subproof is missing.
Theorem. (omega_TransSet)
Proof:
The rest of this subproof is missing.
Theorem. (omega_ordinal)
Proof:
The rest of this subproof is missing.
Theorem. (ordsucc_omega_ordinal)
Proof:
The rest of this subproof is missing.
Definition. We define finite to be λX ⇒ ∃nω, equip X n of type setprop.
Theorem. (nat_finite)
∀n, nat_p nfinite n
Proof:
The rest of this subproof is missing.
Theorem. (finite_ind)
∀p : setprop, p Empty(∀X y, finite Xy Xp Xp (X {y}))∀X, finite Xp X
Proof:
The rest of this subproof is missing.
Theorem. (finite_Empty)
Proof:
The rest of this subproof is missing.
Theorem. (Sing_finite)
∀x, finite {x}
Proof:
The rest of this subproof is missing.
Theorem. (adjoin_finite)
∀X y, finite Xfinite (X {y})
Proof:
The rest of this subproof is missing.
Theorem. (binunion_finite)
∀X, finite X∀Y, finite Yfinite (X Y)
Proof:
The rest of this subproof is missing.
Theorem. (famunion_nat_finite)
∀X : setset, ∀n, nat_p n(∀in, finite (X i))finite (inX i)
Proof:
The rest of this subproof is missing.
Theorem. (Subq_finite)
∀X, finite X∀Y, Y Xfinite Y
Proof:
The rest of this subproof is missing.
Definition. We define infinite to be λA ⇒ ¬ finite A of type setprop.
Beginning of Section InfinitePrimes
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Definition. We define divides_nat to be λm n ⇒ m ω n ω ∃kω, m * k = n of type setsetprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define prime_nat to be λn ⇒ n ω 1 n ∀kω, divides_nat k nk = 1 k = n of type setprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (Pi_nat_divides)
∀f : setset, ∀n, nat_p n(∀in, nat_p (f i))(∀in, divides_nat (f i) (Pi_nat f n))
Proof:
The rest of this subproof is missing.
Definition. We define composite_nat to be λn ⇒ n ω ∃k mω, 1 k 1 m k * m = n of type setprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define primes to be {nω|prime_nat n} of type set.
Proof:
The rest of this subproof is missing.
End of Section InfinitePrimes
Beginning of Section InfiniteRamsey
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (infinite_Finite_Subq_ex)
∀X, infinite X∀n, nat_p n∃YX, equip Y n
Proof:
The rest of this subproof is missing.
Theorem. (infiniteRamsey_lem)
∀X, ∀f g f' : setset, infinite X(∀ZX, infinite Zf Z Z infinite (f Z))(∀ZX, infinite Zg Z Z g Z f Z)f' 0 = f X(∀m, nat_p mf' (ordsucc m) = f (f' m))(∀m, nat_p mf' m X infinite (f' m)) (∀m m'ω, m m'f' m' f' m) (∀m m'ω, g (f' m) = g (f' m')m = m')
Proof:
The rest of this subproof is missing.
Theorem. (infiniteRamsey)
∀c, nat_p c∀n, nat_p n∀X, infinite X∀C : setset, (∀YX, equip Y nC Y c)∃HX, ∃ic, infinite H ∀YH, equip Y nC Y = i
Proof:
The rest of this subproof is missing.
End of Section InfiniteRamsey
Definition. We define Inj1 to be In_rec_i (λX f ⇒ {0} {f x|xX}) of type setset.
Theorem. (Inj1_eq)
∀X : set, Inj1 X = {0} {Inj1 x|xX}
Proof:
The rest of this subproof is missing.
Theorem. (Inj1I1)
∀X : set, 0 Inj1 X
Proof:
The rest of this subproof is missing.
Theorem. (Inj1I2)
∀X x : set, x XInj1 x Inj1 X
Proof:
The rest of this subproof is missing.
Theorem. (Inj1E)
∀X y : set, y Inj1 Xy = 0 ∃xX, y = Inj1 x
Proof:
The rest of this subproof is missing.
Theorem. (Inj1NE1)
∀x : set, Inj1 x 0
Proof:
The rest of this subproof is missing.
Theorem. (Inj1NE2)
∀x : set, Inj1 x {0}
Proof:
The rest of this subproof is missing.
Definition. We define Inj0 to be λX ⇒ {Inj1 x|xX} of type setset.
Theorem. (Inj0I)
∀X x : set, x XInj1 x Inj0 X
Proof:
The rest of this subproof is missing.
Theorem. (Inj0E)
∀X y : set, y Inj0 X∃x : set, x X y = Inj1 x
Proof:
The rest of this subproof is missing.
Definition. We define Unj to be In_rec_i (λX f ⇒ {f x|xX {0}}) of type setset.
Theorem. (Unj_eq)
∀X : set, Unj X = {Unj x|xX {0}}
Proof:
The rest of this subproof is missing.
Theorem. (Unj_Inj1_eq)
∀X : set, Unj (Inj1 X) = X
Proof:
The rest of this subproof is missing.
Theorem. (Inj1_inj)
∀X Y : set, Inj1 X = Inj1 YX = Y
Proof:
The rest of this subproof is missing.
Theorem. (Unj_Inj0_eq)
∀X : set, Unj (Inj0 X) = X
Proof:
The rest of this subproof is missing.
Theorem. (Inj0_inj)
∀X Y : set, Inj0 X = Inj0 YX = Y
Proof:
The rest of this subproof is missing.
Theorem. (Inj0_0)
Proof:
The rest of this subproof is missing.
Theorem. (Inj0_Inj1_neq)
∀X Y : set, Inj0 X Inj1 Y
Proof:
The rest of this subproof is missing.
Definition. We define setsum to be λX Y ⇒ {Inj0 x|xX} {Inj1 y|yY} of type setsetset.
Notation. We use + as an infix operator with priority 450 and which associates to the left corresponding to applying term setsum.
Theorem. (Inj0_setsum)
∀X Y x : set, x XInj0 x X + Y
Proof:
The rest of this subproof is missing.
Theorem. (Inj1_setsum)
∀X Y y : set, y YInj1 y X + Y
Proof:
The rest of this subproof is missing.
Theorem. (setsum_Inj_inv)
∀X Y z : set, z X + Y(∃xX, z = Inj0 x) (∃yY, z = Inj1 y)
Proof:
The rest of this subproof is missing.
Theorem. (Inj0_setsum_0L)
∀X : set, 0 + X = Inj0 X
Proof:
The rest of this subproof is missing.
Theorem. (Inj1_setsum_1L)
∀X : set, 1 + X = Inj1 X
Proof:
The rest of this subproof is missing.
Beginning of Section pair_setsum
Let pair ≝ setsum
Definition. We define proj0 to be λZ ⇒ {Unj z|zZ, ∃x : set, Inj0 x = z} of type setset.
Definition. We define proj1 to be λZ ⇒ {Unj z|zZ, ∃y : set, Inj1 y = z} of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (pairI0)
∀X Y x, x Xpair 0 x pair X Y
Proof:
The rest of this subproof is missing.
Theorem. (pairI1)
∀X Y y, y Ypair 1 y pair X Y
Proof:
The rest of this subproof is missing.
Theorem. (pairE)
∀X Y z, z pair X Y(∃xX, z = pair 0 x) (∃yY, z = pair 1 y)
Proof:
The rest of this subproof is missing.
Theorem. (pairE0)
∀X Y x, pair 0 x pair X Yx X
Proof:
The rest of this subproof is missing.
Theorem. (pairE1)
∀X Y y, pair 1 y pair X Yy Y
Proof:
The rest of this subproof is missing.
Theorem. (proj0I)
∀w u : set, pair 0 u wu proj0 w
Proof:
The rest of this subproof is missing.
Theorem. (proj0E)
∀w u : set, u proj0 wpair 0 u w
Proof:
The rest of this subproof is missing.
Theorem. (proj1I)
∀w u : set, pair 1 u wu proj1 w
Proof:
The rest of this subproof is missing.
Theorem. (proj1E)
∀w u : set, u proj1 wpair 1 u w
Proof:
The rest of this subproof is missing.
Theorem. (proj0_pair_eq)
∀X Y : set, proj0 (pair X Y) = X
Proof:
The rest of this subproof is missing.
Theorem. (proj1_pair_eq)
∀X Y : set, proj1 (pair X Y) = Y
Proof:
The rest of this subproof is missing.
Definition. We define Sigma to be λX Y ⇒ xX{pair x y|yY x} of type set(setset)set.
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Theorem. (Sigma_eta_proj0_proj1)
∀X : set, ∀Y : setset, ∀z(xX, Y x), pair (proj0 z) (proj1 z) = z proj0 z X proj1 z Y (proj0 z)
Proof:
The rest of this subproof is missing.
Theorem. (proj0_Sigma)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)proj0 z X
Proof:
The rest of this subproof is missing.
Theorem. (proj1_Sigma)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)proj1 z Y (proj0 z)
Proof:
The rest of this subproof is missing.
Theorem. (pair_Sigma)
∀X : set, ∀Y : setset, ∀xX, ∀yY x, pair x y xX, Y x
Proof:
The rest of this subproof is missing.
Theorem. (pair_Sigma_E1)
∀X : set, ∀Y : setset, ∀x y : set, pair x y (xX, Y x)y Y x
Proof:
The rest of this subproof is missing.
Theorem. (Sigma_E)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)∃xX, ∃yY x, z = pair x y
Proof:
The rest of this subproof is missing.
Definition. We define setprod to be λX Y : setxX, Y of type setsetset.
Notation. We use as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Let lam : set(setset)setSigma
Definition. We define ap to be λf x ⇒ {proj1 z|zf, ∃y : set, z = pair x y} of type setsetset.
Notation. When x is a set, a term x y is notation for ap x y.
Notation. λ xAB is notation for the set Sigma Ax : set ⇒ B).
Notation. We now use n-tuple notation (a0,...,an-1) for n ≥ 2 for λ i ∈ n . if i = 0 then a0 else ... if i = n-2 then an-2 else an-1.
Theorem. (lamI)
∀X : set, ∀F : setset, ∀xX, ∀yF x, pair x y λxXF x
Proof:
The rest of this subproof is missing.
Theorem. (lamE)
∀X : set, ∀F : setset, ∀z : set, z (λxXF x)∃xX, ∃yF x, z = pair x y
Proof:
The rest of this subproof is missing.
Theorem. (apI)
∀f x y, pair x y fy f x
Proof:
The rest of this subproof is missing.
Theorem. (apE)
∀f x y, y f xpair x y f
Proof:
The rest of this subproof is missing.
Theorem. (beta)
∀X : set, ∀F : setset, ∀x : set, x X(λxXF x) x = F x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (pair_ap_0)
∀x y : set, (pair x y) 0 = x
Proof:
The rest of this subproof is missing.
Theorem. (pair_ap_1)
∀x y : set, (pair x y) 1 = y
Proof:
The rest of this subproof is missing.
Theorem. (ap0_Sigma)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)(z 0) X
Proof:
The rest of this subproof is missing.
Theorem. (ap1_Sigma)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)(z 1) (Y (z 0))
Proof:
The rest of this subproof is missing.
Definition. We define pair_p to be λu : setpair (u 0) (u 1) = u of type setprop.
Theorem. (pair_p_I)
∀x y, pair_p (pair x y)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (tuple_pair)
∀x y : set, pair x y = (x,y)
Proof:
The rest of this subproof is missing.
Definition. We define Pi to be λX Y ⇒ {f𝒫 (xX, (Y x))|∀xX, f x Y x} of type set(setset)set.
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Theorem. (PiI)
∀X : set, ∀Y : setset, ∀f : set, (∀uf, pair_p u u 0 X)(∀xX, f x Y x)f xX, Y x
Proof:
The rest of this subproof is missing.
Theorem. (lam_Pi)
∀X : set, ∀Y : setset, ∀F : setset, (∀xX, F x Y x)(λxXF x) (xX, Y x)
Proof:
The rest of this subproof is missing.
Theorem. (ap_Pi)
∀X : set, ∀Y : setset, ∀f : set, ∀x : set, f (xX, Y x)x Xf x Y x
Proof:
The rest of this subproof is missing.
Definition. We define setexp to be λX Y : setyY, X of type setsetset.
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Theorem. (pair_tuple_fun)
pair = (λx y ⇒ (x,y))
Proof:
The rest of this subproof is missing.
Beginning of Section Tuples
Variable x0 x1 : set
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section Tuples
Theorem. (ReplEq_setprod_ext)
∀X Y, ∀F G : setsetset, (∀xX, ∀yY, F x y = G x y){F (w 0) (w 1)|wX Y} = {G (w 0) (w 1)|wX Y}
Proof:
The rest of this subproof is missing.
Theorem. (lamI2)
∀X, ∀F : setset, ∀xX, ∀yF x, (x,y) λxXF x
Proof:
The rest of this subproof is missing.
Theorem. (tuple_2_Sigma)
∀X : set, ∀Y : setset, ∀xX, ∀yY x, (x,y) xX, Y x
Proof:
The rest of this subproof is missing.
Theorem. (tuple_2_setprod)
∀X : set, ∀Y : set, ∀xX, ∀yY, (x,y) X Y
Proof:
The rest of this subproof is missing.
End of Section pair_setsum
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Notation. We use as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Definition. We define DescrR_i_io_1 to be λR ⇒ Eps_i (λx ⇒ (∃y : setprop, R x y) (∀y z : setprop, R x yR x zy = z)) of type (set(setprop)prop)set.
Definition. We define DescrR_i_io_2 to be λR ⇒ Descr_Vo1 (λy ⇒ R (DescrR_i_io_1 R) y) of type (set(setprop)prop)setprop.
Theorem. (DescrR_i_io_12)
∀R : set(setprop)prop, (∃x, (∃y : setprop, R x y) (∀y z : setprop, R x yR x zy = z))R (DescrR_i_io_1 R) (DescrR_i_io_2 R)
Proof:
The rest of this subproof is missing.
Definition. We define PNoEq_ to be λalpha p q ⇒ ∀betaalpha, p beta q beta of type set(setprop)(setprop)prop.
Theorem. (PNoEq_ref_)
∀alpha, ∀p : setprop, PNoEq_ alpha p p
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_sym_)
∀alpha, ∀p q : setprop, PNoEq_ alpha p qPNoEq_ alpha q p
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_tra_)
∀alpha, ∀p q r : setprop, PNoEq_ alpha p qPNoEq_ alpha q rPNoEq_ alpha p r
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_antimon_)
∀p q : setprop, ∀alpha, ordinal alpha∀betaalpha, PNoEq_ alpha p qPNoEq_ beta p q
Proof:
The rest of this subproof is missing.
Definition. We define PNoLt_ to be λalpha p q ⇒ ∃betaalpha, PNoEq_ beta p q ¬ p beta q beta of type set(setprop)(setprop)prop.
Theorem. (PNoLt_E_)
∀alpha, ∀p q : setprop, PNoLt_ alpha p q∀R : prop, (∀beta, beta alphaPNoEq_ beta p q¬ p betaq betaR)R
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_irref_)
∀alpha, ∀p : setprop, ¬ PNoLt_ alpha p p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_mon_)
∀p q : setprop, ∀alpha, ordinal alpha∀betaalpha, PNoLt_ beta p qPNoLt_ alpha p q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_trichotomy_or_)
∀p q : setprop, ∀alpha, ordinal alphaPNoLt_ alpha p q PNoEq_ alpha p q PNoLt_ alpha q p
Proof:
The rest of this subproof is missing.
Definition. We define PNoLt to be λalpha p beta q ⇒ PNoLt_ (alpha beta) p q alpha beta PNoEq_ alpha p q q alpha beta alpha PNoEq_ beta p q ¬ p beta of type set(setprop)set(setprop)prop.
Theorem. (PNoLtI1)
∀alpha beta, ∀p q : setprop, PNoLt_ (alpha beta) p qPNoLt alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtI2)
∀alpha beta, ∀p q : setprop, alpha betaPNoEq_ alpha p qq alphaPNoLt alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtI3)
∀alpha beta, ∀p q : setprop, beta alphaPNoEq_ beta p q¬ p betaPNoLt alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtE)
∀alpha beta, ∀p q : setprop, PNoLt alpha p beta q∀R : prop, (PNoLt_ (alpha beta) p qR)(alpha betaPNoEq_ alpha p qq alphaR)(beta alphaPNoEq_ beta p q¬ p betaR)R
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_irref)
∀alpha, ∀p : setprop, ¬ PNoLt alpha p alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_trichotomy_or)
∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPNoLt alpha p beta q alpha = beta PNoEq_ alpha p q PNoLt beta q alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtEq_tra)
∀alpha beta, ordinal alphaordinal beta∀p q r : setprop, PNoLt alpha p beta qPNoEq_ beta q rPNoLt alpha p beta r
Proof:
The rest of this subproof is missing.
Theorem. (PNoEqLt_tra)
∀alpha beta, ordinal alphaordinal beta∀p q r : setprop, PNoEq_ alpha p qPNoLt alpha q beta rPNoLt alpha p beta r
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_tra)
∀alpha beta gamma, ordinal alphaordinal betaordinal gamma∀p q r : setprop, PNoLt alpha p beta qPNoLt beta q gamma rPNoLt alpha p gamma r
Proof:
The rest of this subproof is missing.
Definition. We define PNoLe to be λalpha p beta q ⇒ PNoLt alpha p beta q alpha = beta PNoEq_ alpha p q of type set(setprop)set(setprop)prop.
Theorem. (PNoLeI1)
∀alpha beta, ∀p q : setprop, PNoLt alpha p beta qPNoLe alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLeI2)
∀alpha, ∀p q : setprop, PNoEq_ alpha p qPNoLe alpha p alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLe_ref)
∀alpha, ∀p : setprop, PNoLe alpha p alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLe_antisym)
∀alpha beta, ordinal alphaordinal beta∀p q : setprop, PNoLe alpha p beta qPNoLe beta q alpha palpha = beta PNoEq_ alpha p q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtLe_tra)
∀alpha beta gamma, ordinal alphaordinal betaordinal gamma∀p q r : setprop, PNoLt alpha p beta qPNoLe beta q gamma rPNoLt alpha p gamma r
Proof:
The rest of this subproof is missing.
Theorem. (PNoLeLt_tra)
∀alpha beta gamma, ordinal alphaordinal betaordinal gamma∀p q r : setprop, PNoLe alpha p beta qPNoLt beta q gamma rPNoLt alpha p gamma r
Proof:
The rest of this subproof is missing.
Theorem. (PNoEqLe_tra)
∀alpha beta, ordinal alphaordinal beta∀p q r : setprop, PNoEq_ alpha p qPNoLe alpha q beta rPNoLe alpha p beta r
Proof:
The rest of this subproof is missing.
Theorem. (PNoLe_tra)
∀alpha beta gamma, ordinal alphaordinal betaordinal gamma∀p q r : setprop, PNoLe alpha p beta qPNoLe beta q gamma rPNoLe alpha p gamma r
Proof:
The rest of this subproof is missing.
Definition. We define PNo_downc to be λL alpha p ⇒ ∃beta, ordinal beta ∃q : setprop, L beta q PNoLe alpha p beta q of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_upc to be λR alpha p ⇒ ∃beta, ordinal beta ∃q : setprop, R beta q PNoLe beta q alpha p of type (set(setprop)prop)set(setprop)prop.
Theorem. (PNoLe_downc)
∀L : set(setprop)prop, ∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPNo_downc L alpha pPNoLe beta q alpha pPNo_downc L beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_downc_ref)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, L alpha pPNo_downc L alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_upc_ref)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, R alpha pPNo_upc R alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLe_upc)
∀R : set(setprop)prop, ∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPNo_upc R alpha pPNoLe alpha p beta qPNo_upc R beta q
Proof:
The rest of this subproof is missing.
Definition. We define PNoLt_pwise to be λL R ⇒ ∀gamma, ordinal gamma∀p : setprop, L gamma p∀delta, ordinal delta∀q : setprop, R delta qPNoLt gamma p delta q of type (set(setprop)prop)(set(setprop)prop)prop.
Theorem. (PNoLt_pwise_downc_upc)
∀L R : set(setprop)prop, PNoLt_pwise L RPNoLt_pwise (PNo_downc L) (PNo_upc R)
Proof:
The rest of this subproof is missing.
Definition. We define PNo_rel_strict_upperbd to be λL alpha p ⇒ ∀betaalpha, ∀q : setprop, PNo_downc L beta qPNoLt beta q alpha p of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_rel_strict_lowerbd to be λR alpha p ⇒ ∀betaalpha, ∀q : setprop, PNo_upc R beta qPNoLt alpha p beta q of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_rel_strict_imv to be λL R alpha p ⇒ PNo_rel_strict_upperbd L alpha p PNo_rel_strict_lowerbd R alpha p of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Theorem. (PNoEq_rel_strict_upperbd)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_rel_strict_upperbd L alpha pPNo_rel_strict_upperbd L alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_strict_upperbd_antimon)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, ∀betaalpha, PNo_rel_strict_upperbd L alpha pPNo_rel_strict_upperbd L beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_rel_strict_lowerbd)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_rel_strict_lowerbd R alpha pPNo_rel_strict_lowerbd R alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_strict_lowerbd_antimon)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, ∀betaalpha, PNo_rel_strict_lowerbd R alpha pPNo_rel_strict_lowerbd R beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_rel_strict_imv)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_rel_strict_imv L R alpha pPNo_rel_strict_imv L R alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_strict_imv_antimon)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, ∀betaalpha, PNo_rel_strict_imv L R alpha pPNo_rel_strict_imv L R beta p
Proof:
The rest of this subproof is missing.
Definition. We define PNo_rel_strict_uniq_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R alpha p ∀q : setprop, PNo_rel_strict_imv L R alpha qPNoEq_ alpha p q of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Definition. We define PNo_rel_strict_split_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta delta alpha) PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta delta = alpha) of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Theorem. (PNo_extend0_eq)
∀alpha, ∀p : setprop, PNoEq_ alpha p (λdelta ⇒ p delta delta alpha)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_extend1_eq)
∀alpha, ∀p : setprop, PNoEq_ alpha p (λdelta ⇒ p delta delta = alpha)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_imv_ex)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alpha(∃p : setprop, PNo_rel_strict_uniq_imv L R alpha p) (∃taualpha, ∃p : setprop, PNo_rel_strict_split_imv L R tau p)
Proof:
The rest of this subproof is missing.
Definition. We define PNo_lenbdd to be λalpha L ⇒ ∀beta, ∀p : setprop, L beta pbeta alpha of type set(set(setprop)prop)prop.
Theorem. (PNo_lenbdd_strict_imv_extend0)
∀L R : set(setprop)prop, ∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∀p : setprop, PNo_rel_strict_imv L R alpha pPNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta delta alpha)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_lenbdd_strict_imv_extend1)
∀L R : set(setprop)prop, ∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∀p : setprop, PNo_rel_strict_imv L R alpha pPNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta delta = alpha)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_lenbdd_strict_imv_split)
∀L R : set(setprop)prop, ∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∀p : setprop, PNo_rel_strict_imv L R alpha pPNo_rel_strict_split_imv L R alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_imv_bdd_ex)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∃betaordsucc alpha, ∃p : setprop, PNo_rel_strict_split_imv L R beta p
Proof:
The rest of this subproof is missing.
Definition. We define PNo_strict_upperbd to be λL alpha p ⇒ ∀beta, ordinal beta∀q : setprop, L beta qPNoLt beta q alpha p of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_strict_lowerbd to be λR alpha p ⇒ ∀beta, ordinal beta∀q : setprop, R beta qPNoLt alpha p beta q of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_strict_imv to be λL R alpha p ⇒ PNo_strict_upperbd L alpha p PNo_strict_lowerbd R alpha p of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Theorem. (PNoEq_strict_upperbd)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_strict_upperbd L alpha pPNo_strict_upperbd L alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_strict_lowerbd)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_strict_lowerbd R alpha pPNo_strict_lowerbd R alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_strict_imv)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_strict_imv L R alpha pPNo_strict_imv L R alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_strict_upperbd_imp_rel_strict_upperbd)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀betaordsucc alpha, ∀p : setprop, PNo_strict_upperbd L alpha pPNo_rel_strict_upperbd L beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_strict_lowerbd_imp_rel_strict_lowerbd)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀betaordsucc alpha, ∀p : setprop, PNo_strict_lowerbd R alpha pPNo_rel_strict_lowerbd R beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_strict_imv_imp_rel_strict_imv)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀betaordsucc alpha, ∀p : setprop, PNo_strict_imv L R alpha pPNo_rel_strict_imv L R beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_split_imv_imp_strict_imv)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, PNo_rel_strict_split_imv L R alpha pPNo_strict_imv L R alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_lenbdd_strict_imv_ex)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∃betaordsucc alpha, ∃p : setprop, PNo_strict_imv L R beta p
Proof:
The rest of this subproof is missing.
Definition. We define PNo_least_rep to be λL R beta p ⇒ ordinal beta PNo_strict_imv L R beta p ∀gammabeta, ∀q : setprop, ¬ PNo_strict_imv L R gamma q of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Definition. We define PNo_least_rep2 to be λL R beta p ⇒ PNo_least_rep L R beta p ∀x, x beta¬ p x of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Theorem. (PNo_strict_imv_pred_eq)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alpha∀p q : setprop, PNo_least_rep L R alpha pPNo_strict_imv L R alpha q∀betaalpha, p beta q beta
Proof:
The rest of this subproof is missing.
Theorem. (PNo_lenbdd_least_rep2_exuniq2)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∃beta, (∃p : setprop, PNo_least_rep2 L R beta p) (∀p q : setprop, PNo_least_rep2 L R beta pPNo_least_rep2 L R beta qp = q)
Proof:
The rest of this subproof is missing.
Definition. We define PNo_bd to be λL R ⇒ DescrR_i_io_1 (PNo_least_rep2 L R) of type (set(setprop)prop)(set(setprop)prop)set.
Definition. We define PNo_pred to be λL R ⇒ DescrR_i_io_2 (PNo_least_rep2 L R) of type (set(setprop)prop)(set(setprop)prop)setprop.
Theorem. (PNo_bd_pred_lem)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha RPNo_least_rep2 L R (PNo_bd L R) (PNo_pred L R)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_bd_pred)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha RPNo_least_rep L R (PNo_bd L R) (PNo_pred L R)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_bd_In)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha RPNo_bd L R ordsucc alpha
Proof:
The rest of this subproof is missing.
Beginning of Section TaggedSets
Let tag : setsetλalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (tagged_notin_ordinal)
∀alpha y, ordinal alpha(y ') alpha
Proof:
The rest of this subproof is missing.
Theorem. (tagged_eqE_Subq)
∀alpha beta, ordinal alphaalpha ' = beta 'alpha beta
Proof:
The rest of this subproof is missing.
Theorem. (tagged_eqE_eq)
∀alpha beta, ordinal alphaordinal betaalpha ' = beta 'alpha = beta
Proof:
The rest of this subproof is missing.
Theorem. (tagged_ReplE)
∀alpha beta, ordinal alphaordinal betabeta ' {gamma '|gammaalpha}beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_notin_tagged_Repl)
∀alpha Y, ordinal alphaalpha {y '|yY}
Proof:
The rest of this subproof is missing.
Definition. We define SNoElts_ to be λalpha ⇒ alpha {beta '|betaalpha} of type setset.
Theorem. (SNoElts_mon)
∀alpha beta, alpha betaSNoElts_ alpha SNoElts_ beta
Proof:
The rest of this subproof is missing.
Definition. We define SNo_ to be λalpha x ⇒ x SNoElts_ alpha ∀betaalpha, exactly1of2 (beta ' x) (beta x) of type setsetprop.
Definition. We define PSNo to be λalpha p ⇒ {betaalpha|p beta} {beta '|betaalpha, ¬ p beta} of type set(setprop)set.
Theorem. (PNoEq_PSNo)
∀alpha, ordinal alpha∀p : setprop, PNoEq_ alpha (λbeta ⇒ beta PSNo alpha p) p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_PSNo)
∀alpha, ordinal alpha∀p : setprop, SNo_ alpha (PSNo alpha p)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_PSNo_eta_)
∀alpha x, ordinal alphaSNo_ alpha xx = PSNo alpha (λbeta ⇒ beta x)
Proof:
The rest of this subproof is missing.
Definition. We define SNo to be λx ⇒ ∃alpha, ordinal alpha SNo_ alpha x of type setprop.
Theorem. (SNo_SNo)
∀alpha, ordinal alpha∀z, SNo_ alpha zSNo z
Proof:
The rest of this subproof is missing.
Definition. We define SNoLev to be λx ⇒ Eps_i (λalpha ⇒ ordinal alpha SNo_ alpha x) of type setset.
Theorem. (SNoLev_uniq_Subq)
∀x alpha beta, ordinal alphaordinal betaSNo_ alpha xSNo_ beta xalpha beta
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_uniq)
∀x alpha beta, ordinal alphaordinal betaSNo_ alpha xSNo_ beta xalpha = beta
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_)
∀x, SNo xSNo_ (SNoLev x) x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_PSNo_eta)
∀x, SNo xx = PSNo (SNoLev x) (λbeta ⇒ beta x)
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_PSNo)
∀alpha, ordinal alpha∀p : setprop, SNoLev (PSNo alpha p) = alpha
Proof:
The rest of this subproof is missing.
Theorem. (SNo_Subq)
∀x y, SNo xSNo ySNoLev x SNoLev y(∀alphaSNoLev x, alpha x alpha y)x y
Proof:
The rest of this subproof is missing.
Definition. We define SNoEq_ to be λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta x) (λbeta ⇒ beta y) of type setsetsetprop.
Theorem. (SNoEq_I)
∀alpha x y, (∀betaalpha, beta x beta y)SNoEq_ alpha x y
Proof:
The rest of this subproof is missing.
Theorem. (SNo_eq)
∀x y, SNo xSNo ySNoLev x = SNoLev ySNoEq_ (SNoLev x) x yx = y
Proof:
The rest of this subproof is missing.
End of Section TaggedSets
Definition. We define SNoLt to be λx y ⇒ PNoLt (SNoLev x) (λbeta ⇒ beta x) (SNoLev y) (λbeta ⇒ beta y) of type setsetprop.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Definition. We define SNoLe to be λx y ⇒ PNoLe (SNoLev x) (λbeta ⇒ beta x) (SNoLev y) (λbeta ⇒ beta y) of type setsetprop.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (SNoLtLe)
∀x y, x < yx y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLeE)
∀x y, SNo xSNo yx yx < y x = y
Proof:
The rest of this subproof is missing.
Theorem. (SNoEq_sym_)
∀alpha x y, SNoEq_ alpha x ySNoEq_ alpha y x
Proof:
The rest of this subproof is missing.
Theorem. (SNoEq_tra_)
∀alpha x y z, SNoEq_ alpha x ySNoEq_ alpha y zSNoEq_ alpha x z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtE)
∀x y, SNo xSNo yx < y∀p : prop, (∀z, SNo zSNoLev z SNoLev x SNoLev ySNoEq_ (SNoLev z) z xSNoEq_ (SNoLev z) z yx < zz < ySNoLev z xSNoLev z yp)(SNoLev x SNoLev ySNoEq_ (SNoLev x) x ySNoLev x yp)(SNoLev y SNoLev xSNoEq_ (SNoLev y) x ySNoLev y xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtI2)
∀x y, SNoLev x SNoLev ySNoEq_ (SNoLev x) x ySNoLev x yx < y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtI3)
∀x y, SNoLev y SNoLev xSNoEq_ (SNoLev y) x ySNoLev y xx < y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_irref)
∀x, ¬ SNoLt x x
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_trichotomy_or)
∀x y, SNo xSNo yx < y x = y y < x
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_trichotomy_or_impred)
∀x y, SNo xSNo y∀p : prop, (x < yp)(x = yp)(y < xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_tra)
∀x y z, SNo xSNo ySNo zx < yy < zx < z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLe_ref)
∀x, SNoLe x x
Proof:
The rest of this subproof is missing.
Theorem. (SNoLe_antisym)
∀x y, SNo xSNo yx yy xx = y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtLe_tra)
∀x y z, SNo xSNo ySNo zx < yy zx < z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLeLt_tra)
∀x y z, SNo xSNo ySNo zx yy < zx < z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLe_tra)
∀x y z, SNo xSNo ySNo zx yy zx z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtLe_or)
∀x y, SNo xSNo yx < y y x
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_PSNo_PNoLt)
∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPSNo alpha p < PSNo beta qPNoLt alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_SNoLt_PSNo)
∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPNoLt alpha p beta qPSNo alpha p < PSNo beta q
Proof:
The rest of this subproof is missing.
Definition. We define SNoCut to be λL R ⇒ PSNo (PNo_bd (λalpha p ⇒ ordinal alpha PSNo alpha p L) (λalpha p ⇒ ordinal alpha PSNo alpha p R)) (PNo_pred (λalpha p ⇒ ordinal alpha PSNo alpha p L) (λalpha p ⇒ ordinal alpha PSNo alpha p R)) of type setsetset.
Definition. We define SNoCutP to be λL R ⇒ (∀xL, SNo x) (∀yR, SNo y) (∀xL, ∀yR, x < y) of type setsetprop.
Theorem. (SNoCutP_SNoCut)
∀L R, SNoCutP L RSNo (SNoCut L R) SNoLev (SNoCut L R) ordsucc ((xLordsucc (SNoLev x)) (yRordsucc (SNoLev y))) (∀xL, x < SNoCut L R) (∀yR, SNoCut L R < y) (∀z, SNo z(∀xL, x < z)(∀yR, z < y)SNoLev (SNoCut L R) SNoLev z SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z)
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_impred)
∀L R, SNoCutP L R∀p : prop, (SNo (SNoCut L R)SNoLev (SNoCut L R) ordsucc ((xLordsucc (SNoLev x)) (yRordsucc (SNoLev y)))(∀xL, x < SNoCut L R)(∀yR, SNoCut L R < y)(∀z, SNo z(∀xL, x < z)(∀yR, z < y)SNoLev (SNoCut L R) SNoLev z SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z)p)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_L_0)
∀L, (∀xL, SNo x)SNoCutP L 0
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_0_0)
Proof:
The rest of this subproof is missing.
Definition. We define SNoS_ to be λalpha ⇒ {x𝒫 (SNoElts_ alpha)|∃betaalpha, SNo_ beta x} of type setset.
Theorem. (SNoS_E)
∀alpha, ordinal alpha∀xSNoS_ alpha, ∃betaalpha, SNo_ beta x
Proof:
The rest of this subproof is missing.
Beginning of Section TaggedSets2
Let tag : setsetλalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Theorem. (SNoS_I)
∀alpha, ordinal alpha∀x, ∀betaalpha, SNo_ beta xx SNoS_ alpha
Proof:
The rest of this subproof is missing.
Theorem. (SNoS_I2)
∀x y, SNo xSNo ySNoLev x SNoLev yx SNoS_ (SNoLev y)
Proof:
The rest of this subproof is missing.
Theorem. (SNoS_Subq)
∀alpha beta, ordinal alphaordinal betaalpha betaSNoS_ alpha SNoS_ beta
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_uniq2)
∀alpha, ordinal alpha∀x, SNo_ alpha xSNoLev x = alpha
Proof:
The rest of this subproof is missing.
Theorem. (SNoS_E2)
∀alpha, ordinal alpha∀xSNoS_ alpha, ∀p : prop, (SNoLev x alphaordinal (SNoLev x)SNo xSNo_ (SNoLev x) xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoS_In_neq)
∀w, SNo w∀xSNoS_ (SNoLev w), x w
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define SNoL to be λz ⇒ {xSNoS_ (SNoLev z)|x < z} of type setset.
Definition. We define SNoR to be λz ⇒ {ySNoS_ (SNoLev z)|z < y} of type setset.
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_E)
∀x, SNo x∀wSNoL x, ∀p : prop, (SNo wSNoLev w SNoLev xw < xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoR_E)
∀x, SNo x∀zSNoR x, ∀p : prop, (SNo zSNoLev z SNoLev xx < zp)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_SNoS)
∀x, SNo x∀wSNoL x, w SNoS_ (SNoLev x)
Proof:
The rest of this subproof is missing.
Theorem. (SNoR_SNoS)
∀x, SNo x∀zSNoR x, z SNoS_ (SNoLev x)
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_I)
∀z, SNo z∀x, SNo xSNoLev x SNoLev zx < zx SNoL z
Proof:
The rest of this subproof is missing.
Theorem. (SNoR_I)
∀z, SNo z∀y, SNo ySNoLev y SNoLev zz < yy SNoR z
Proof:
The rest of this subproof is missing.
Theorem. (SNo_eta)
∀z, SNo zz = SNoCut (SNoL z) (SNoR z)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_L)
∀L R, SNoCutP L R∀xL, x < SNoCut L R
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_R)
∀L R, SNoCutP L R∀yR, SNoCut L R < y
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_fst)
∀L R, SNoCutP L R∀z, SNo z(∀xL, x < z)(∀yR, z < y)SNoLev (SNoCut L R) SNoLev z SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z
Proof:
The rest of this subproof is missing.
Theorem. (SNoCut_Le)
∀L1 R1 L2 R2, SNoCutP L1 R1SNoCutP L2 R2(∀wL1, w < SNoCut L2 R2)(∀zR2, SNoCut L1 R1 < z)SNoCut L1 R1 SNoCut L2 R2
Proof:
The rest of this subproof is missing.
Theorem. (SNoCut_ext)
∀L1 R1 L2 R2, SNoCutP L1 R1SNoCutP L2 R2(∀wL1, w < SNoCut L2 R2)(∀zR1, SNoCut L2 R2 < z)(∀wL2, w < SNoCut L1 R1)(∀zR2, SNoCut L1 R1 < z)SNoCut L1 R1 = SNoCut L2 R2
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_SNoL_or_SNoR_impred)
∀x y, SNo xSNo yx < y∀p : prop, (∀zSNoL y, z SNoR xp)(x SNoL yp)(y SNoR xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_or_SNoR_impred)
∀x y, SNo xSNo y∀p : prop, (x = yp)(∀zSNoL y, z SNoR xp)(x SNoL yp)(y SNoR xp)(∀zSNoR y, z SNoL xp)(x SNoR yp)(y SNoL xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_SNoCutP_ex)
∀L R, SNoCutP L R∀wSNoL (SNoCut L R), ∃w'L, w w'
Proof:
The rest of this subproof is missing.
Theorem. (SNoR_SNoCutP_ex)
∀L R, SNoCutP L R∀zSNoR (SNoCut L R), ∃z'R, z' z
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNo_)
∀alpha, ordinal alphaSNo_ alpha alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNo)
∀alpha, ordinal alphaSNo alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoLev)
∀alpha, ordinal alphaSNoLev alpha = alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoLev_max)
∀alpha, ordinal alpha∀z, SNo zSNoLev z alphaz < alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoL)
∀alpha, ordinal alphaSNoL alpha = SNoS_ alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoR)
∀alpha, ordinal alphaSNoR alpha = Empty
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_In_SNoLt)
∀alpha, ordinal alpha∀betaalpha, beta < alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoLev_max_2)
∀alpha, ordinal alpha∀z, SNo zSNoLev z ordsucc alphaz alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_Subq_SNoLe)
∀alpha beta, ordinal alphaordinal betaalpha betaalpha beta
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoLt_In)
∀alpha beta, ordinal alphaordinal betaalpha < betaalpha beta
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_max_SNoLev)
∀x, SNo x(∀ySNoS_ (SNoLev x), y < x)SNoLev x = x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_max_ordinal)
∀x, SNo x(∀ySNoS_ (SNoLev x), y < x)ordinal x
Proof:
The rest of this subproof is missing.
Theorem. (pos_low_eq_one)
∀x, SNo x0 < xSNoLev x 1x = 1
Proof:
The rest of this subproof is missing.
Definition. We define SNo_extend0 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta x delta SNoLev x) of type setset.
Definition. We define SNo_extend1 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta x delta = SNoLev x) of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_0_eq_0)
∀x, SNo xSNoLev x = 0x = 0
Proof:
The rest of this subproof is missing.
Definition. We define eps_ to be λn ⇒ {0} {(ordsucc m) '|mn} of type setset.
Theorem. (eps_ordinal_In_eq_0)
∀n alpha, ordinal alphaalpha eps_ nalpha = 0
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_pos_eps_Lt)
∀n, nat_p n∀xSNoS_ (ordsucc n), 0 < xeps_ n < x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_pos_eps_Le)
∀n, nat_p n∀xSNoS_ (ordsucc (ordsucc n)), 0 < xeps_ n x
Proof:
The rest of this subproof is missing.
Theorem. (eps_SNo_eq)
∀n, nat_p n∀xSNoS_ (ordsucc n), 0 < xSNoEq_ (SNoLev x) (eps_ n) x∃mn, x = eps_ m
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section TaggedSets2
Theorem. (SNo_etaE)
∀z, SNo z∀p : prop, (∀L R, SNoCutP L R(∀xL, SNoLev x SNoLev z)(∀yR, SNoLev y SNoLev z)z = SNoCut L Rp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_ind)
∀P : setprop, (∀L R, SNoCutP L R(∀xL, P x)(∀yR, P y)P (SNoCut L R))∀z, SNo zP z
Proof:
The rest of this subproof is missing.
Beginning of Section SurrealRecI
Variable F : set(setset)set
Let default : setEps_i (λ_ ⇒ True)
Let G : set(setsetset)setsetλalpha g ⇒ If_ii (ordinal alpha) (λz : setif z SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) (λz : setdefault)
Definition. We define SNo_rec_i to be λz ⇒ In_rec_ii G (SNoLev z) z of type setset.
Hypothesis Fr : ∀z, SNo z∀g h : setset, (∀wSNoS_ (SNoLev z), g w = h w)F z g = F z h
Proof:
The rest of this subproof is missing.
End of Section SurrealRecI
Beginning of Section SurrealRecII
Variable F : set(set(setset))(setset)
Let default : (setset)Descr_ii (λ_ ⇒ True)
Let G : set(setset(setset))set(setset)λalpha g ⇒ If_iii (ordinal alpha) (λz : setIf_ii (z SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : setdefault)
Definition. We define SNo_rec_ii to be λz ⇒ In_rec_iii G (SNoLev z) z of type set(setset).
Hypothesis Fr : ∀z, SNo z∀g h : set(setset), (∀wSNoS_ (SNoLev z), g w = h w)F z g = F z h
Proof:
The rest of this subproof is missing.
End of Section SurrealRecII
Beginning of Section SurrealRec2
Variable F : setset(setsetset)set
Let G : set(setsetset)set(setset)setλw f z g ⇒ F w z (λx y ⇒ if x = w then g y else f x y)
Let H : set(setsetset)setsetλw f z ⇒ if SNo z then SNo_rec_i (G w f) z else Empty
Definition. We define SNo_rec2 to be SNo_rec_ii H of type setsetset.
Hypothesis Fr : ∀w, SNo w∀z, SNo z∀g h : setsetset, (∀xSNoS_ (SNoLev w), ∀y, SNo yg x y = h x y)(∀ySNoS_ (SNoLev z), g w y = h w y)F w z g = F w z h
Theorem. (SNo_rec2_G_prop)
∀w, SNo w∀f k : setsetset, (∀xSNoS_ (SNoLev w), f x = k x)∀z, SNo z∀g h : setset, (∀uSNoS_ (SNoLev z), g u = h u)G w f z g = G w k z h
Proof:
The rest of this subproof is missing.
Theorem. (SNo_rec2_eq_1)
∀w, SNo w∀f : setsetset, ∀z, SNo zSNo_rec_i (G w f) z = G w f z (SNo_rec_i (G w f))
Proof:
The rest of this subproof is missing.
Theorem. (SNo_rec2_eq)
∀w, SNo w∀z, SNo zSNo_rec2 w z = F w z SNo_rec2
Proof:
The rest of this subproof is missing.
End of Section SurrealRec2
Theorem. (SNo_ordinal_ind)
∀P : setprop, (∀alpha, ordinal alpha∀xSNoS_ alpha, P x)(∀x, SNo xP x)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_ordinal_ind2)
∀P : setsetprop, (∀alpha, ordinal alpha∀beta, ordinal beta∀xSNoS_ alpha, ∀ySNoS_ beta, P x y)(∀x y, SNo xSNo yP x y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_ordinal_ind3)
∀P : setsetsetprop, (∀alpha, ordinal alpha∀beta, ordinal beta∀gamma, ordinal gamma∀xSNoS_ alpha, ∀ySNoS_ beta, ∀zSNoS_ gamma, P x y z)(∀x y z, SNo xSNo ySNo zP x y z)
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_ind)
∀P : setprop, (∀x, SNo x(∀wSNoS_ (SNoLev x), P w)P x)(∀x, SNo xP x)
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_ind2)
∀P : setsetprop, (∀x y, SNo xSNo y(∀wSNoS_ (SNoLev x), P w y)(∀zSNoS_ (SNoLev y), P x z)(∀wSNoS_ (SNoLev x), ∀zSNoS_ (SNoLev y), P w z)P x y)∀x y, SNo xSNo yP x y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_ind3)
∀P : setsetsetprop, (∀x y z, SNo xSNo ySNo z(∀uSNoS_ (SNoLev x), P u y z)(∀vSNoS_ (SNoLev y), P x v z)(∀wSNoS_ (SNoLev z), P x y w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), P u v z)(∀uSNoS_ (SNoLev x), ∀wSNoS_ (SNoLev z), P u y w)(∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), P x v w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), P u v w)P x y z)∀x y z, SNo xSNo ySNo zP x y z
Proof:
The rest of this subproof is missing.
Theorem. (SNo_omega)
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_0_1)
0 < 1
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_0_2)
0 < 2
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_1_2)
1 < 2
Proof:
The rest of this subproof is missing.
Theorem. (restr_SNo_)
∀x, SNo x∀alphaSNoLev x, SNo_ alpha (x SNoElts_ alpha)
Proof:
The rest of this subproof is missing.
Theorem. (restr_SNo)
∀x, SNo x∀alphaSNoLev x, SNo (x SNoElts_ alpha)
Proof:
The rest of this subproof is missing.
Theorem. (restr_SNoLev)
∀x, SNo x∀alphaSNoLev x, SNoLev (x SNoElts_ alpha) = alpha
Proof:
The rest of this subproof is missing.
Theorem. (restr_SNoEq)
∀x, SNo x∀alphaSNoLev x, SNoEq_ alpha (x SNoElts_ alpha) x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_extend0_restr_eq)
∀x, SNo xx = SNo_extend0 x SNoElts_ (SNoLev x)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_extend1_restr_eq)
∀x, SNo xx = SNo_extend1 x SNoElts_ (SNoLev x)
Proof:
The rest of this subproof is missing.
Beginning of Section SurrealMinus
Definition. We define minus_SNo to be SNo_rec_i (λx m ⇒ SNoCut {m z|zSNoR x} {m w|wSNoL x}) of type setset.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_prop1)
∀x, SNo xSNo (- x) (∀uSNoL x, - x < - u) (∀uSNoR x, - u < - x) SNoCutP {- z|zSNoR x} {- w|wSNoL x}
Proof:
The rest of this subproof is missing.
Theorem. (SNo_minus_SNo)
∀x, SNo xSNo (- x)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Lt_contra)
∀x y, SNo xSNo yx < y- y < - x
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Le_contra)
∀x y, SNo xSNo yx y- y - x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Lev_lem1)
∀alpha, ordinal alpha∀xSNoS_ alpha, SNoLev (- x) SNoLev x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_invol)
∀x, SNo x- - x = x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_SNo_)
∀alpha, ordinal alpha∀x, SNo_ alpha xSNo_ alpha (- x)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_SNoS_)
∀alpha, ordinal alpha∀x, x SNoS_ alpha- x SNoS_ alpha
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNoCut_eq_lem)
∀v, SNo v∀L R, SNoCutP L Rv = SNoCut L R- v = SNoCut {- z|zR} {- w|wL}
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Lt_contra1)
∀x y, SNo xSNo y- x < y- y < x
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Lt_contra2)
∀x y, SNo xSNo yx < - yy < - x
Proof:
The rest of this subproof is missing.
Theorem. (mordinal_SNoLev_min_2)
∀alpha, ordinal alpha∀z, SNo zSNoLev z ordsucc alpha- alpha z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SurrealMinus
Beginning of Section SurrealAdd
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Definition. We define add_SNo to be SNo_rec2 (λx y a ⇒ SNoCut ({a w y|wSNoL x} {a x w|wSNoL y}) ({a z y|zSNoR x} {a x z|zSNoR y})) of type setsetset.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Theorem. (add_SNo_eq)
∀x, SNo x∀y, SNo yx + y = SNoCut ({w + y|wSNoL x} {x + w|wSNoL y}) ({z + y|zSNoR x} {x + z|zSNoR y})
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_prop1)
∀x y, SNo xSNo ySNo (x + y) (∀uSNoL x, u + y < x + y) (∀uSNoR x, x + y < u + y) (∀uSNoL y, x + u < x + y) (∀uSNoR y, x + y < x + u) SNoCutP ({w + y|wSNoL x} {x + w|wSNoL y}) ({z + y|zSNoR x} {x + z|zSNoR y})
Proof:
The rest of this subproof is missing.
Theorem. (SNo_add_SNo)
∀x y, SNo xSNo ySNo (x + y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_add_SNo_3)
∀x y z, SNo xSNo ySNo zSNo (x + y + z)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_add_SNo_3c)
∀x y z, SNo xSNo ySNo zSNo (x + y + - z)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_add_SNo_4)
∀x y z w, SNo xSNo ySNo zSNo wSNo (x + y + z + w)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt1)
∀x y z, SNo xSNo ySNo zx < zx + y < z + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Le1)
∀x y z, SNo xSNo ySNo zx zx + y z + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt2)
∀x y z, SNo xSNo ySNo zy < zx + y < x + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Le2)
∀x y z, SNo xSNo ySNo zy zx + y x + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt3a)
∀x y z w, SNo xSNo ySNo zSNo wx < zy wx + y < z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt3b)
∀x y z w, SNo xSNo ySNo zSNo wx zy < wx + y < z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt3)
∀x y z w, SNo xSNo ySNo zSNo wx < zy < wx + y < z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Le3)
∀x y z w, SNo xSNo ySNo zSNo wx zy wx + y z + w
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_com)
∀x y, SNo xSNo yx + y = y + x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_0L)
∀x, SNo x0 + x = x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_0R)
∀x, SNo xx + 0 = x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_eq)
∀alpha, ordinal alpha∀beta, ordinal betaalpha + beta = SNoCut ({x + beta|xSNoS_ alpha} {alpha + x|xSNoS_ beta}) Empty
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_ordinal)
∀alpha, ordinal alpha∀beta, ordinal betaordinal (alpha + beta)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_SL)
∀alpha, ordinal alpha∀beta, ordinal betaordsucc alpha + beta = ordsucc (alpha + beta)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_SR)
∀alpha, ordinal alpha∀beta, ordinal betaalpha + ordsucc beta = ordsucc (alpha + beta)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_InL)
∀alpha, ordinal alpha∀beta, ordinal beta∀gammaalpha, gamma + beta alpha + beta
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_InR)
∀alpha, ordinal alpha∀beta, ordinal beta∀gammabeta, alpha + gamma alpha + beta
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_SNoL_interpolate)
∀x y, SNo xSNo y∀uSNoL (x + y), (∃vSNoL x, u v + y) (∃vSNoL y, u x + v)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_SNoR_interpolate)
∀x y, SNo xSNo y∀uSNoR (x + y), (∃vSNoR x, v + y u) (∃vSNoR y, x + v u)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_assoc)
∀x y z, SNo xSNo ySNo zx + (y + z) = (x + y) + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_R2)
∀x y, SNo xSNo y(x + y) + - y = x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_R2')
∀x y, SNo xSNo y(x + - y) + y = x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_L2)
∀x y, SNo xSNo y- x + (x + y) = y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_L2')
∀x y, SNo xSNo yx + (- x + y) = y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_cancel_L)
∀x y z, SNo xSNo ySNo zx + y = x + zy = z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_cancel_R)
∀x y z, SNo xSNo ySNo zx + y = z + yx = z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_add_SNo_distr)
∀x y, SNo xSNo y- (x + y) = (- x) + (- y)
Proof:
The rest of this subproof is missing.
Theorem. (minus_add_SNo_distr_3)
∀x y z, SNo xSNo ySNo z- (x + y + z) = - x + - y + - z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lev_bd)
∀x y, SNo xSNo ySNoLev (x + y) SNoLev x + SNoLev y
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt1_cancel)
∀x y z, SNo xSNo ySNo zx + y < z + yx < z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt2_cancel)
∀x y z, SNo xSNo ySNo zx + y < x + zy < z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Le1_cancel)
∀x y z, SNo xSNo ySNo zx + y z + yx z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_assoc_4)
∀x y z w, SNo xSNo ySNo zSNo wx + y + z + w = (x + y + z) + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_com_3_0_1)
∀x y z, SNo xSNo ySNo zx + y + z = y + x + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_com_3b_1_2)
∀x y z, SNo xSNo ySNo z(x + y) + z = (x + z) + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_com_4_inner_mid)
∀x y z w, SNo xSNo ySNo zSNo w(x + y) + (z + w) = (x + z) + (y + w)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_rotate_3_1)
∀x y z, SNo xSNo ySNo zx + y + z = z + x + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_rotate_4_1)
∀x y z w, SNo xSNo ySNo zSNo wx + y + z + w = w + x + y + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_rotate_5_1)
∀x y z w v, SNo xSNo ySNo zSNo wSNo vx + y + z + w + v = v + x + y + z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_rotate_5_2)
∀x y z w v, SNo xSNo ySNo zSNo wSNo vx + y + z + w + v = w + v + x + y + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_SNo_prop2)
∀x y, SNo xSNo yx + - x + y = y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_SNo_prop3)
∀x y z w, SNo xSNo ySNo zSNo w(x + y + z) + (- z + w) = x + y + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_SNo_prop5)
∀x y z w, SNo xSNo ySNo zSNo w(x + y + - z) + (z + w) = x + y + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt1)
∀x y z, SNo xSNo ySNo zx + - y < zx < z + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt2)
∀x y z, SNo xSNo ySNo zz < x + - yz + y < x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt1b)
∀x y z, SNo xSNo ySNo zx < z + yx + - y < z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt2b)
∀x y z, SNo xSNo ySNo zz + y < xz < x + - y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt1b3)
∀x y z w, SNo xSNo ySNo zSNo wx + y < w + zx + y + - z < w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt2b3)
∀x y z w, SNo xSNo ySNo zSNo ww + z < x + yw < x + y + - z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt_lem)
∀x y z u v w, SNo xSNo ySNo zSNo uSNo vSNo wx + y + w < u + v + zx + y + - z < u + v + - w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le2)
∀x y z, SNo xSNo ySNo zz x + - yz + y x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le2b)
∀x y z, SNo xSNo ySNo zz + y xz x + - y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop2)
∀x y z w u v, SNo xSNo ySNo zSNo wSNo uSNo vx + u < z + vy + v < w + ux + y < z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop3a)
∀x y z w u a, SNo xSNo ySNo zSNo wSNo uSNo ax + z < w + ay + a < ux + y + z < w + u
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop3b)
∀x y w u v a, SNo xSNo ySNo wSNo uSNo vSNo ax + a < w + vy < a + ux + y < w + u + v
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop3c)
∀x y z w u a b c, SNo xSNo ySNo zSNo wSNo uSNo aSNo bSNo cx + a < b + cy + c < ub + z < w + ax + y + z < w + u
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop3d)
∀x y w u v a b c, SNo xSNo ySNo wSNo uSNo vSNo aSNo bSNo cx + a < b + vy < c + ub + c < w + ax + y < w + u + v
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_SNo_eq)
∀alpha, ordinal alphaordsucc alpha = 1 + alpha
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_3a_2b)
∀x y z w u, SNo xSNo ySNo zSNo wSNo u(x + y + z) + (w + u) = (u + y + z) + (w + x)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_eps_Lt)
∀x, SNo x∀nω, x < x + eps_ n
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_eps_Lt')
∀x y, SNo xSNo y∀nω, x < yx < y + eps_ n
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_minus_pos)
∀x y, SNo xSNo yx < y0 < y + - x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_omega_In_cases)
∀m, ∀nω, ∀k, nat_p km n + km n m + - n k
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt4)
∀x y z w u v, SNo xSNo ySNo zSNo wSNo uSNo vx < wy < uz < vx + y + z < w + u + v
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_3_3_3_Lt1)
∀x y z w u, SNo xSNo ySNo zSNo wSNo ux + y < z + wx + y + u < z + w + u
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_3_2_3_Lt1)
∀x y z w u, SNo xSNo ySNo zSNo wSNo uy + x < z + wx + u + y < z + w + u
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt12b3)
∀x y z w u v, SNo xSNo ySNo zSNo wSNo uSNo vx + y + v < w + u + zx + y + - z < w + u + - v
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le1b)
∀x y z, SNo xSNo ySNo zx z + yx + - y z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le1b3)
∀x y z w, SNo xSNo ySNo zSNo wx + y w + zx + y + - z w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le12b3)
∀x y z w u v, SNo xSNo ySNo zSNo wSNo uSNo vx + y + v w + u + zx + y + - z w + u + - v
Proof:
The rest of this subproof is missing.
End of Section SurrealAdd
Beginning of Section SurrealAbs
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define abs_SNo to be λx ⇒ if 0 x then x else - x of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (pos_abs_SNo)
∀x, 0 < xabs_SNo x = x
Proof:
The rest of this subproof is missing.
Theorem. (neg_abs_SNo)
∀x, SNo xx < 0abs_SNo x = - x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (abs_SNo_dist_swap)
∀x y, SNo xSNo yabs_SNo (x + - y) = abs_SNo (y + - x)
Proof:
The rest of this subproof is missing.
End of Section SurrealAbs
Beginning of Section SurrealMul
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Definition. We define mul_SNo to be SNo_rec2 (λx y m ⇒ SNoCut ({m (w 0) y + m x (w 1) + - m (w 0) (w 1)|wSNoL x SNoL y} {m (z 0) y + m x (z 1) + - m (z 0) (z 1)|zSNoR x SNoR y}) ({m (w 0) y + m x (w 1) + - m (w 0) (w 1)|wSNoL x SNoR y} {m (z 0) y + m x (z 1) + - m (z 0) (z 1)|zSNoR x SNoL y})) of type setsetset.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Theorem. (mul_SNo_eq)
∀x, SNo x∀y, SNo yx * y = SNoCut ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|wSNoL x SNoL y} {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|zSNoR x SNoR y}) ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|wSNoL x SNoR y} {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|zSNoR x SNoL y})
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_eq_2)
∀x y, SNo xSNo y∀p : prop, (∀L R, (∀u, u L(∀q : prop, (∀w0SNoL x, ∀w1SNoL y, u = w0 * y + x * w1 + - w0 * w1q)(∀z0SNoR x, ∀z1SNoR y, u = z0 * y + x * z1 + - z0 * z1q)q))(∀w0SNoL x, ∀w1SNoL y, w0 * y + x * w1 + - w0 * w1 L)(∀z0SNoR x, ∀z1SNoR y, z0 * y + x * z1 + - z0 * z1 L)(∀u, u R(∀q : prop, (∀w0SNoL x, ∀z1SNoR y, u = w0 * y + x * z1 + - w0 * z1q)(∀z0SNoR x, ∀w1SNoL y, u = z0 * y + x * w1 + - z0 * w1q)q))(∀w0SNoL x, ∀z1SNoR y, w0 * y + x * z1 + - w0 * z1 R)(∀z0SNoR x, ∀w1SNoL y, z0 * y + x * w1 + - z0 * w1 R)x * y = SNoCut L Rp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_prop_1)
∀x, SNo x∀y, SNo y∀p : prop, (SNo (x * y)(∀uSNoL x, ∀vSNoL y, u * y + x * v < x * y + u * v)(∀uSNoR x, ∀vSNoR y, u * y + x * v < x * y + u * v)(∀uSNoL x, ∀vSNoR y, x * y + u * v < u * y + x * v)(∀uSNoR x, ∀vSNoL y, x * y + u * v < u * y + x * v)p)p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_mul_SNo)
∀x y, SNo xSNo ySNo (x * y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_mul_SNo_lem)
∀x y u v, SNo xSNo ySNo uSNo vSNo (u * y + x * v + - (u * v))
Proof:
The rest of this subproof is missing.
Theorem. (SNo_mul_SNo_3)
∀x y z, SNo xSNo ySNo zSNo (x * y * z)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_eq_3)
∀x y, SNo xSNo y∀p : prop, (∀L R, SNoCutP L R(∀u, u L(∀q : prop, (∀w0SNoL x, ∀w1SNoL y, u = w0 * y + x * w1 + - w0 * w1q)(∀z0SNoR x, ∀z1SNoR y, u = z0 * y + x * z1 + - z0 * z1q)q))(∀w0SNoL x, ∀w1SNoL y, w0 * y + x * w1 + - w0 * w1 L)(∀z0SNoR x, ∀z1SNoR y, z0 * y + x * z1 + - z0 * z1 L)(∀u, u R(∀q : prop, (∀w0SNoL x, ∀z1SNoR y, u = w0 * y + x * z1 + - w0 * z1q)(∀z0SNoR x, ∀w1SNoL y, u = z0 * y + x * w1 + - z0 * w1q)q))(∀w0SNoL x, ∀z1SNoR y, w0 * y + x * z1 + - w0 * z1 R)(∀z0SNoR x, ∀w1SNoL y, z0 * y + x * w1 + - z0 * w1 R)x * y = SNoCut L Rp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Lt)
∀x y u v, SNo xSNo ySNo uSNo vu < xv < yu * y + x * v < x * y + u * v
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Le)
∀x y u v, SNo xSNo ySNo uSNo vu xv yu * y + x * v x * y + u * v
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoL_interpolate)
∀x y, SNo xSNo y∀uSNoL (x * y), (∃vSNoL x, ∃wSNoL y, u + v * w v * y + x * w) (∃vSNoR x, ∃wSNoR y, u + v * w v * y + x * w)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoL_interpolate_impred)
∀x y, SNo xSNo y∀uSNoL (x * y), ∀p : prop, (∀vSNoL x, ∀wSNoL y, u + v * w v * y + x * wp)(∀vSNoR x, ∀wSNoR y, u + v * w v * y + x * wp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoR_interpolate)
∀x y, SNo xSNo y∀uSNoR (x * y), (∃vSNoL x, ∃wSNoR y, v * y + x * w u + v * w) (∃vSNoR x, ∃wSNoL y, v * y + x * w u + v * w)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoR_interpolate_impred)
∀x y, SNo xSNo y∀uSNoR (x * y), ∀p : prop, (∀vSNoL x, ∀wSNoR y, v * y + x * w u + v * wp)(∀vSNoR x, ∀wSNoL y, v * y + x * w u + v * wp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Subq_lem)
∀x y X Y Z W, ∀U U', (∀u, u U(∀q : prop, (∀w0X, ∀w1Y, u = w0 * y + x * w1 + - w0 * w1q)(∀z0Z, ∀z1W, u = z0 * y + x * z1 + - z0 * z1q)q))(∀w0X, ∀w1Y, w0 * y + x * w1 + - w0 * w1 U')(∀w0Z, ∀w1W, w0 * y + x * w1 + - w0 * w1 U')U U'
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_zeroR)
∀x, SNo xx * 0 = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_oneR)
∀x, SNo xx * 1 = x
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_com)
∀x y, SNo xSNo yx * y = y * x
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_minus_distrL)
∀x y, SNo xSNo y(- x) * y = - x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_minus_distrR)
∀x y, SNo xSNo yx * (- y) = - (x * y)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_distrR)
∀x y z, SNo xSNo ySNo z(x + y) * z = x * z + y * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_distrL)
∀x y z, SNo xSNo ySNo zx * (y + z) = x * y + x * z
Proof:
The rest of this subproof is missing.
Beginning of Section mul_SNo_assoc_lems
Variable M : setsetset
Hypothesis DL : ∀x y z, SNo xSNo ySNo zx * (y + z) = x * y + x * z
Hypothesis DR : ∀x y z, SNo xSNo ySNo z(x + y) * z = x * z + y * z
Hypothesis IL : ∀x y, SNo xSNo y∀uSNoL (x * y), ∀p : prop, (∀vSNoL x, ∀wSNoL y, u + v * w v * y + x * wp)(∀vSNoR x, ∀wSNoR y, u + v * w v * y + x * wp)p
Hypothesis IR : ∀x y, SNo xSNo y∀uSNoR (x * y), ∀p : prop, (∀vSNoL x, ∀wSNoR y, v * y + x * w u + v * wp)(∀vSNoR x, ∀wSNoL y, v * y + x * w u + v * wp)p
Hypothesis M_Lt : ∀x y u v, SNo xSNo ySNo uSNo vu < xv < yu * y + x * v < x * y + u * v
Hypothesis M_Le : ∀x y u v, SNo xSNo ySNo uSNo vu xv yu * y + x * v x * y + u * v
Theorem. (mul_SNo_assoc_lem1)
∀x y z, SNo xSNo ySNo z(∀uSNoS_ (SNoLev x), u * (y * z) = (u * y) * z)(∀vSNoS_ (SNoLev y), x * (v * z) = (x * v) * z)(∀wSNoS_ (SNoLev z), x * (y * w) = (x * y) * w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), u * (v * z) = (u * v) * z)(∀uSNoS_ (SNoLev x), ∀wSNoS_ (SNoLev z), u * (y * w) = (u * y) * w)(∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), x * (v * w) = (x * v) * w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), u * (v * w) = (u * v) * w)∀L, (∀uL, ∀q : prop, (∀vSNoL x, ∀wSNoL (y * z), u = v * (y * z) + x * w + - v * wq)(∀vSNoR x, ∀wSNoR (y * z), u = v * (y * z) + x * w + - v * wq)q)∀uL, u < (x * y) * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_assoc_lem2)
∀x y z, SNo xSNo ySNo z(∀uSNoS_ (SNoLev x), u * (y * z) = (u * y) * z)(∀vSNoS_ (SNoLev y), x * (v * z) = (x * v) * z)(∀wSNoS_ (SNoLev z), x * (y * w) = (x * y) * w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), u * (v * z) = (u * v) * z)(∀uSNoS_ (SNoLev x), ∀wSNoS_ (SNoLev z), u * (y * w) = (u * y) * w)(∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), x * (v * w) = (x * v) * w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), u * (v * w) = (u * v) * w)∀R, (∀uR, ∀q : prop, (∀vSNoL x, ∀wSNoR (y * z), u = v * (y * z) + x * w + - v * wq)(∀vSNoR x, ∀wSNoL (y * z), u = v * (y * z) + x * w + - v * wq)q)∀uR, (x * y) * z < u
Proof:
The rest of this subproof is missing.
End of Section mul_SNo_assoc_lems
Theorem. (mul_SNo_assoc)
∀x y z, SNo xSNo ySNo zx * (y * z) = (x * y) * z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_zeroL)
∀x, SNo x0 * x = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_oneL)
∀x, SNo x1 * x = x
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_rotate_3_1)
∀x y z, SNo xSNo ySNo zx * y * z = z * x * y
Proof:
The rest of this subproof is missing.
Theorem. (pos_mul_SNo_Lt)
∀x y z, SNo x0 < xSNo ySNo zy < zx * y < x * z
Proof:
The rest of this subproof is missing.
Theorem. (nonneg_mul_SNo_Le)
∀x y z, SNo x0 xSNo ySNo zy zx * y x * z
Proof:
The rest of this subproof is missing.
Theorem. (neg_mul_SNo_Lt)
∀x y z, SNo xx < 0SNo ySNo zz < yx * y < x * z
Proof:
The rest of this subproof is missing.
Theorem. (pos_mul_SNo_Lt')
∀x y z, SNo xSNo ySNo z0 < zx < yx * z < y * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Lt1_pos_Lt)
∀x y, SNo xSNo yx < 10 < yx * y < y
Proof:
The rest of this subproof is missing.
Theorem. (nonneg_mul_SNo_Le')
∀x y z, SNo xSNo ySNo z0 zx yx * z y * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Le1_nonneg_Le)
∀x y, SNo xSNo yx 10 yx * y y
Proof:
The rest of this subproof is missing.
Theorem. (pos_mul_SNo_Lt2)
∀x y z w, SNo xSNo ySNo zSNo w0 < x0 < yx < zy < wx * y < z * w
Proof:
The rest of this subproof is missing.
Theorem. (nonneg_mul_SNo_Le2)
∀x y z w, SNo xSNo ySNo zSNo w0 x0 yx zy wx * y z * w
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_pos_pos)
∀x y, SNo xSNo y0 < x0 < y0 < x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_pos_neg)
∀x y, SNo xSNo y0 < xy < 0x * y < 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_neg_pos)
∀x y, SNo xSNo yx < 00 < yx * y < 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_neg_neg)
∀x y, SNo xSNo yx < 0y < 00 < x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonneg_nonneg)
∀x y, SNo xSNo y0 x0 y0 x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonpos_pos)
∀x y, SNo xSNo yx 00 < yx * y 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonpos_neg)
∀x y, SNo xSNo yx 0y < 00 x * y
Proof:
The rest of this subproof is missing.
Theorem. (nonpos_mul_SNo_Le)
∀x y z, SNo xx 0SNo ySNo zz yx * y x * z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_pos_sqr_uniq)
∀x y, SNo xSNo y0 < x0 < yx * x = y * yx = y
Proof:
The rest of this subproof is missing.
Theorem. (SNo_nonneg_sqr_uniq)
∀x y, SNo xSNo y0 x0 yx * x = y * yx = y
Proof:
The rest of this subproof is missing.
Theorem. (SNo_foil)
∀x y z w, SNo xSNo ySNo zSNo w(x + y) * (z + w) = x * z + x * w + y * z + y * w
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_minus_minus)
∀x y, SNo xSNo y(- x) * (- y) = x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_com_3_0_1)
∀x y z, SNo xSNo ySNo zx * y * z = y * x * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_com_3b_1_2)
∀x y z, SNo xSNo ySNo z(x * y) * z = (x * z) * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_com_4_inner_mid)
∀x y z w, SNo xSNo ySNo zSNo w(x * y) * (z * w) = (x * z) * (y * w)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_foil_mm)
∀x y z w, SNo xSNo ySNo zSNo w(x + - y) * (z + - w) = x * z + - x * w + - y * z + y * w
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonzero_cancel)
∀x y z, SNo xx 0SNo ySNo zx * y = x * zy = z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNoCutP_lem)
∀Lx Rx Ly Ry x y, SNoCutP Lx RxSNoCutP Ly Ryx = SNoCut Lx Rxy = SNoCut Ly RySNoCutP ({w 0 * y + x * w 1 + - w 0 * w 1|wLx Ly} {z 0 * y + x * z 1 + - z 0 * z 1|zRx Ry}) ({w 0 * y + x * w 1 + - w 0 * w 1|wLx Ry} {z 0 * y + x * z 1 + - z 0 * z 1|zRx Ly}) x * y = SNoCut ({w 0 * y + x * w 1 + - w 0 * w 1|wLx Ly} {z 0 * y + x * z 1 + - z 0 * z 1|zRx Ry}) ({w 0 * y + x * w 1 + - w 0 * w 1|wLx Ry} {z 0 * y + x * z 1 + - z 0 * z 1|zRx Ly}) ∀q : prop, (∀LxLy' RxRy' LxRy' RxLy', (∀uLxLy', ∀p : prop, (∀wLx, ∀w'Ly, SNo wSNo w'w < xw' < yu = w * y + x * w' + - w * w'p)p)(∀wLx, ∀w'Ly, w * y + x * w' + - w * w' LxLy')(∀uRxRy', ∀p : prop, (∀zRx, ∀z'Ry, SNo zSNo z'x < zy < z'u = z * y + x * z' + - z * z'p)p)(∀zRx, ∀z'Ry, z * y + x * z' + - z * z' RxRy')(∀uLxRy', ∀p : prop, (∀wLx, ∀zRy, SNo wSNo zw < xy < zu = w * y + x * z + - w * zp)p)(∀wLx, ∀zRy, w * y + x * z + - w * z LxRy')(∀uRxLy', ∀p : prop, (∀zRx, ∀wLy, SNo zSNo wx < zw < yu = z * y + x * w + - z * wp)p)(∀zRx, ∀wLy, z * y + x * w + - z * w RxLy')SNoCutP (LxLy' RxRy') (LxRy' RxLy')x * y = SNoCut (LxLy' RxRy') (LxRy' RxLy')q)q
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNoCut_abs)
∀Lx Rx Ly Ry x y, SNoCutP Lx RxSNoCutP Ly Ryx = SNoCut Lx Rxy = SNoCut Ly Ry∀q : prop, (∀LxLy' RxRy' LxRy' RxLy', (∀uLxLy', ∀p : prop, (∀wLx, ∀w'Ly, SNo wSNo w'w < xw' < yu = w * y + x * w' + - w * w'p)p)(∀wLx, ∀w'Ly, w * y + x * w' + - w * w' LxLy')(∀uRxRy', ∀p : prop, (∀zRx, ∀z'Ry, SNo zSNo z'x < zy < z'u = z * y + x * z' + - z * z'p)p)(∀zRx, ∀z'Ry, z * y + x * z' + - z * z' RxRy')(∀uLxRy', ∀p : prop, (∀wLx, ∀zRy, SNo wSNo zw < xy < zu = w * y + x * z + - w * zp)p)(∀wLx, ∀zRy, w * y + x * z + - w * z LxRy')(∀uRxLy', ∀p : prop, (∀zRx, ∀wLy, SNo zSNo wx < zw < yu = z * y + x * w + - z * wp)p)(∀zRx, ∀wLy, z * y + x * w + - z * w RxLy')SNoCutP (LxLy' RxRy') (LxRy' RxLy')x * y = SNoCut (LxLy' RxRy') (LxRy' RxLy')q)q
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoCut_SNoL_interpolate)
∀Lx Rx Ly Ry, SNoCutP Lx RxSNoCutP Ly Ry∀x y, x = SNoCut Lx Rxy = SNoCut Ly Ry∀uSNoL (x * y), (∃vLx, ∃wLy, u + v * w v * y + x * w) (∃vRx, ∃wRy, u + v * w v * y + x * w)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoCut_SNoL_interpolate_impred)
∀Lx Rx Ly Ry, SNoCutP Lx RxSNoCutP Ly Ry∀x y, x = SNoCut Lx Rxy = SNoCut Ly Ry∀uSNoL (x * y), ∀p : prop, (∀vLx, ∀wLy, u + v * w v * y + x * wp)(∀vRx, ∀wRy, u + v * w v * y + x * wp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoCut_SNoR_interpolate)
∀Lx Rx Ly Ry, SNoCutP Lx RxSNoCutP Ly Ry∀x y, x = SNoCut Lx Rxy = SNoCut Ly Ry∀uSNoR (x * y), (∃vLx, ∃wRy, v * y + x * w u + v * w) (∃vRx, ∃wLy, v * y + x * w u + v * w)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoCut_SNoR_interpolate_impred)
∀Lx Rx Ly Ry, SNoCutP Lx RxSNoCutP Ly Ry∀x y, x = SNoCut Lx Rxy = SNoCut Ly Ry∀uSNoR (x * y), ∀p : prop, (∀vLx, ∀wRy, v * y + x * w u + v * wp)(∀vRx, ∀wLy, v * y + x * w u + v * wp)p
Proof:
The rest of this subproof is missing.
Theorem. (nonpos_nonneg_0)
∀m nω, m = - nm = 0 n = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_minus_SNo_distrR)
∀x y, SNo xSNo yx * (- y) = - (x * y)
Proof:
The rest of this subproof is missing.
End of Section SurrealMul
Beginning of Section Int
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define int to be ω {- n|nω} of type set.
Theorem. (int_SNo_cases)
∀p : setprop, (∀nω, p n)(∀nω, p (- n))∀xint, p x
Proof:
The rest of this subproof is missing.
Theorem. (int_3_cases)
∀nint, ∀p : prop, (∀mω, n = - ordsucc mp)(n = 0p)(∀mω, n = ordsucc mp)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section Int
Beginning of Section BezoutAndGCD
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (quotient_remainder_nat)
∀nω {0}, ∀m, nat_p m∃qω, ∃rn, m = q * n + r
Proof:
The rest of this subproof is missing.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (mul_SNo_nonpos_nonneg)
∀x y, SNo xSNo yx 00 yx * y 0
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_pos)
∀alpha, ordinal alpha0 < ordsucc alpha
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define divides_int to be λm n ⇒ m int n int ∃kint, m * k = n of type setsetprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (divides_int_mul_SNo)
∀m n m' n', divides_int m m'divides_int n n'divides_int (m * n) (m' * n')
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (divides_int_pos_Le)
∀m n, divides_int m n0 < nm n
Proof:
The rest of this subproof is missing.
Definition. We define gcd_reln to be λm n d ⇒ divides_int d m divides_int d n ∀d', divides_int d' mdivides_int d' nd' d of type setsetsetprop.
Theorem. (gcd_reln_uniq)
∀a b c d, gcd_reln a b cgcd_reln a b dc = d
Proof:
The rest of this subproof is missing.
Definition. We define int_lin_comb to be λa b c ⇒ a int b int c int ∃m nint, m * a + n * b = c of type setsetsetprop.
Theorem. (int_lin_comb_I)
∀a b cint, (∃m nint, m * a + n * b = c)int_lin_comb a b c
Proof:
The rest of this subproof is missing.
Theorem. (int_lin_comb_E)
∀a b c, int_lin_comb a b c∀p : prop, (a intb intc int∀m nint, m * a + n * b = cp)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (int_lin_comb_E4)
∀a b c, int_lin_comb a b c∀p : prop, (∀m nint, m * a + n * b = cp)p
Proof:
The rest of this subproof is missing.
Theorem. (least_pos_int_lin_comb_ex)
∀a bint, ¬ (a = 0 b = 0)∃c, int_lin_comb a b c 0 < c ∀c', int_lin_comb a b c'0 < c'c c'
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (least_pos_int_lin_comb_divides_int)
∀a b d, int_lin_comb a b d0 < d(∀c, int_lin_comb a b c0 < cd c)divides_int d a
Proof:
The rest of this subproof is missing.
Theorem. (least_pos_int_lin_comb_gcd)
∀a b d, int_lin_comb a b d0 < d(∀c, int_lin_comb a b c0 < cd c)gcd_reln a b d
Proof:
The rest of this subproof is missing.
Theorem. (BezoutThm)
∀a bint, ¬ (a = 0 b = 0)∀d, gcd_reln a b d int_lin_comb a b d 0 < d ∀d', int_lin_comb a b d'0 < d'd d'
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (gcd_sym)
∀m n d, gcd_reln m n dgcd_reln n m d
Proof:
The rest of this subproof is missing.
Theorem. (gcd_minus)
∀m n d, gcd_reln m n dgcd_reln m (- n) d
Proof:
The rest of this subproof is missing.
Theorem. (euclidean_algorithm_prop_1)
∀m n d, n intgcd_reln m (n + - m) dgcd_reln m n d
Proof:
The rest of this subproof is missing.
Theorem. (euclidean_algorithm)
(∀mω {0}, gcd_reln m m m) (∀mω {0}, gcd_reln 0 m m) (∀mω {0}, gcd_reln m 0 m) (∀m nω, m < n∀d, gcd_reln m (n + - m) dgcd_reln m n d) (∀m nω, n < m∀d, gcd_reln n m dgcd_reln m n d) (∀mω, ∀nint, n < 0∀d, gcd_reln m (- n) dgcd_reln m n d) (∀m nint, m < 0∀d, gcd_reln (- m) n dgcd_reln m n d)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section BezoutAndGCD
Beginning of Section PrimeFactorization
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Proof:
The rest of this subproof is missing.
Definition. We define Pi_SNo to be λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type (setset)setset.
Theorem. (Pi_SNo_0)
∀f : setset, Pi_SNo f 0 = 1
Proof:
The rest of this subproof is missing.
Theorem. (Pi_SNo_S)
∀f : setset, ∀n, nat_p nPi_SNo f (ordsucc n) = Pi_SNo f n * f n
Proof:
The rest of this subproof is missing.
Theorem. (Pi_SNo_In_omega)
∀f : setset, ∀n, nat_p n(∀in, f i ω)Pi_SNo f n ω
Proof:
The rest of this subproof is missing.
Theorem. (Pi_SNo_In_int)
∀f : setset, ∀n, nat_p n(∀in, f i int)Pi_SNo f n int
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (Euclid_lemma_Pi_SNo)
∀f : setset, ∀p, prime_nat p∀n, nat_p n(∀in, f i int)divides_int p (Pi_SNo f n)∃in, divides_int p (f i)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (Pi_SNo_divides)
∀f : setset, ∀n, nat_p n(∀in, f i ω)(∀in, divides_nat (f i) (Pi_SNo f n))
Proof:
The rest of this subproof is missing.
Definition. We define nonincrfinseq to be λA n f ⇒ ∀in, A (f i) ∀ji, f i f j of type (setprop)set(setset)prop.
Theorem. (Pi_SNo_eq)
∀f g : setset, ∀m, nat_p m(∀im, f i = g i)Pi_SNo f m = Pi_SNo g m
Proof:
The rest of this subproof is missing.
Theorem. (prime_factorization_ex_uniq)
∀n, nat_p n0 n∃kω, ∃f : setset, nonincrfinseq prime_nat k f Pi_SNo f k = n ∀k'ω, ∀f' : setset, nonincrfinseq prime_nat k' f'Pi_SNo f' k' = nk' = k ∀ik, f' i = f i
Proof:
The rest of this subproof is missing.
End of Section PrimeFactorization
Beginning of Section SurrealExp
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define exp_SNo_nat to be λn m : setnat_primrec 1 (λ_ r ⇒ n * r) m of type setsetset.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Theorem. (exp_SNo_nat_0)
∀x, SNo xx ^ 0 = 1
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_S)
∀x, SNo x∀n, nat_p nx ^ (ordsucc n) = x * x ^ n
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_1)
∀x, SNo xx ^ 1 = x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_exp_SNo_nat)
∀x, SNo x∀n, nat_p nSNo (x ^ n)
Proof:
The rest of this subproof is missing.
Theorem. (nat_exp_SNo_nat)
∀x, nat_p x∀n, nat_p nnat_p (x ^ n)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (double_eps_1)
∀x y z, SNo xSNo ySNo zx + x = y + zx = eps_ 1 * (y + z)
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_1_bd)
∀x, SNo x1 x∀n, nat_p n1 x ^ n
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_2_bd)
∀n, nat_p nn < 2 ^ n
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_mul_add)
∀x, SNo x∀m, nat_p m∀n, nat_p nx ^ m * x ^ n = x ^ (m + n)
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_mul_add')
∀x, SNo x∀m nω, x ^ m * x ^ n = x ^ (m + n)
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_pos)
∀x, SNo x0 < x∀n, nat_p n0 < x ^ n
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SurrealExp
Beginning of Section SNoMaxMin
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Definition. We define SNo_max_of to be λX x ⇒ x X SNo x ∀yX, SNo yy x of type setsetprop.
Definition. We define SNo_min_of to be λX x ⇒ x X SNo x ∀yX, SNo yx y of type setsetprop.
Theorem. (minus_SNo_max_min)
∀X y, (∀xX, SNo x)SNo_max_of X ySNo_min_of {- x|xX} (- y)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_max_min')
∀X y, (∀xX, SNo x)SNo_max_of {- x|xX} ySNo_min_of X (- y)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_min_max)
∀X y, (∀xX, SNo x)SNo_min_of X ySNo_max_of {- x|xX} (- y)
Proof:
The rest of this subproof is missing.
Theorem. (double_SNo_max_1)
∀x y, SNo xSNo_max_of (SNoL x) y∀z, SNo zx < zy + z < x + x∃wSNoR z, y + w = x + x
Proof:
The rest of this subproof is missing.
Theorem. (double_SNo_min_1)
∀x y, SNo xSNo_min_of (SNoR x) y∀z, SNo zz < xx + x < y + z∃wSNoL z, y + w = x + x
Proof:
The rest of this subproof is missing.
Theorem. (finite_max_exists)
∀X, (∀xX, SNo x)finite XX 0∃x, SNo_max_of X x
Proof:
The rest of this subproof is missing.
Theorem. (finite_min_exists)
∀X, (∀xX, SNo x)finite XX 0∃x, SNo_min_of X x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SNoMaxMin
Beginning of Section DiadicRationals
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Proof:
The rest of this subproof is missing.
Definition. We define diadic_rational_p to be λx ⇒ ∃kω, ∃mint, x = eps_ k * m of type setprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section DiadicRationals
Beginning of Section SurrealDiv
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Definition. We define SNoL_pos to be λx ⇒ {wSNoL x|0 < w} of type setset.
Theorem. (SNo_recip_pos_pos)
∀x xi, SNo xSNo xi0 < xx * xi = 10 < xi
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recip_lem1)
∀x x' x'i y y', SNo x0 < xx' SNoL_pos xSNo x'ix' * x'i = 1SNo yx * y < 1SNo y'1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i1 < x * y'
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recip_lem2)
∀x x' x'i y y', SNo x0 < xx' SNoL_pos xSNo x'ix' * x'i = 1SNo y1 < x * ySNo y'1 + - x * y' = (1 + - x * y) * (x' + - x) * x'ix * y' < 1
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recip_lem3)
∀x x' x'i y y', SNo x0 < xx' SNoR xSNo x'ix' * x'i = 1SNo yx * y < 1SNo y'1 + - x * y' = (1 + - x * y) * (x' + - x) * x'ix * y' < 1
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recip_lem4)
∀x x' x'i y y', SNo x0 < xx' SNoR xSNo x'ix' * x'i = 1SNo y1 < x * ySNo y'1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i1 < x * y'
Proof:
The rest of this subproof is missing.
Definition. We define SNo_recipauxset to be λY x X g ⇒ yY{(1 + (x' + - x) * y) * g x'|x'X} of type setsetset(setset)set.
Theorem. (SNo_recipauxset_I)
∀Y x X, ∀g : setset, ∀yY, ∀x'X, (1 + (x' + - x) * y) * g x' SNo_recipauxset Y x X g
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipauxset_E)
∀Y x X, ∀g : setset, ∀zSNo_recipauxset Y x X g, ∀p : prop, (∀yY, ∀x'X, z = (1 + (x' + - x) * y) * g x'p)p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipauxset_ext)
∀Y x X, ∀g h : setset, (∀x'X, g x' = h x')SNo_recipauxset Y x X g = SNo_recipauxset Y x X h
Proof:
The rest of this subproof is missing.
Definition. We define SNo_recipaux to be λx g ⇒ nat_primrec ({0},0) (λk p ⇒ (p 0 SNo_recipauxset (p 0) x (SNoR x) g SNo_recipauxset (p 1) x (SNoL_pos x) g,p 1 SNo_recipauxset (p 0) x (SNoL_pos x) g SNo_recipauxset (p 1) x (SNoR x) g)) of type set(setset)setset.
Theorem. (SNo_recipaux_0)
∀x, ∀g : setset, SNo_recipaux x g 0 = ({0},0)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipaux_S)
∀x, ∀g : setset, ∀n, nat_p nSNo_recipaux x g (ordsucc n) = (SNo_recipaux x g n 0 SNo_recipauxset (SNo_recipaux x g n 0) x (SNoR x) g SNo_recipauxset (SNo_recipaux x g n 1) x (SNoL_pos x) g,SNo_recipaux x g n 1 SNo_recipauxset (SNo_recipaux x g n 0) x (SNoL_pos x) g SNo_recipauxset (SNo_recipaux x g n 1) x (SNoR x) g)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipaux_lem1)
∀x, SNo x0 < x∀g : setset, (∀x'SNoS_ (SNoLev x), 0 < x'SNo (g x') x' * g x' = 1)∀k, nat_p k(∀ySNo_recipaux x g k 0, SNo y x * y < 1) (∀ySNo_recipaux x g k 1, SNo y 1 < x * y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipaux_lem2)
∀x, SNo x0 < x∀g : setset, (∀x'SNoS_ (SNoLev x), 0 < x'SNo (g x') x' * g x' = 1)SNoCutP (kωSNo_recipaux x g k 0) (kωSNo_recipaux x g k 1)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipaux_ext)
∀x, SNo x∀g h : setset, (∀x'SNoS_ (SNoLev x), g x' = h x')∀k, nat_p kSNo_recipaux x g k = SNo_recipaux x h k
Proof:
The rest of this subproof is missing.
Beginning of Section recip_SNo_pos
Let G : set(setset)setλx g ⇒ SNoCut (kωSNo_recipaux x g k 0) (kωSNo_recipaux x g k 1)
Definition. We define recip_SNo_pos to be SNo_rec_i G of type setset.
Proof:
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Proof:
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Proof:
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Proof:
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Proof:
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Proof:
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Proof:
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Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section recip_SNo_pos
Definition. We define recip_SNo to be λx ⇒ if 0 < x then recip_SNo_pos x else if x < 0 then - recip_SNo_pos (- x) else 0 of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
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Proof:
The rest of this subproof is missing.
Theorem. (recip_SNo_invR)
∀x, SNo xx 0x * recip_SNo x = 1
Proof:
The rest of this subproof is missing.
Theorem. (recip_SNo_invL)
∀x, SNo xx 0recip_SNo x * x = 1
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonzero_cancel_L)
∀x y z, SNo xx 0SNo ySNo zx * y = x * zy = z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define div_SNo to be λx y ⇒ x * recip_SNo y of type setsetset.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Theorem. (SNo_div_SNo)
∀x y, SNo xSNo ySNo (x :/: y)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_invL)
∀x y, SNo xSNo yy 0(x :/: y) * y = x
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_invR)
∀x y, SNo xSNo yy 0y * (x :/: y) = x
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_R)
∀x y z, SNo xSNo ySNo z(x :/: y) * z = (x * z) :/: y
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_L)
∀x y z, SNo xSNo ySNo zz * (x :/: y) = (z * x) :/: y
Proof:
The rest of this subproof is missing.
Theorem. (div_mul_SNo_invL)
∀x y, SNo xSNo yy 0(x * y) :/: y = x
Proof:
The rest of this subproof is missing.
Theorem. (div_div_SNo)
∀x y z, SNo xSNo ySNo z(x :/: y) :/: z = x :/: (y * z)
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_both)
∀x y z w, SNo xSNo ySNo zSNo w(x :/: y) * (z :/: w) = (x * z) :/: (y * w)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_pos)
∀x y, SNo xSNo y0 < x0 < y0 < x :/: y
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_neg_pos)
∀x y, SNo xSNo yx < 00 < yx :/: y < 0
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_LtL)
∀x y z, SNo xSNo ySNo z0 < yx < z * yx :/: y < z
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_LtR)
∀x y z, SNo xSNo ySNo z0 < yz * y < xz < x :/: y
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_LtL')
∀x y z, SNo xSNo ySNo z0 < yx :/: y < zx < z * y
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_LtR')
∀x y z, SNo xSNo ySNo z0 < yz < x :/: yz * y < x
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_nonzero_eq)
∀x y z, SNo xSNo ySNo zy 0x = y * zx :/: y = z
Proof:
The rest of this subproof is missing.
End of Section SurrealDiv
Beginning of Section Reals
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define real to be {xSNoS_ (ordsucc ω)|x ω x - ω (∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)} of type set.
Theorem. (real_I)
∀xSNoS_ (ordsucc ω), x ωx - ω(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)x real
Proof:
The rest of this subproof is missing.
Theorem. (real_E)
∀xreal, ∀p : prop, (SNo xSNoLev x ordsucc ωx SNoS_ (ordsucc ω)- ω < xx < ω(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)p)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (real_SNoCut_SNoS_omega)
∀L RSNoS_ ω, SNoCutP L RL 0R 0(∀wL, ∃w'L, w < w')(∀zR, ∃z'R, z' < z)SNoCut L R real
Proof:
The rest of this subproof is missing.
Theorem. (real_SNoCut)
∀L Rreal, SNoCutP L RL 0R 0(∀wL, ∃w'L, w < w')(∀zR, ∃z'R, z' < z)SNoCut L R real
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_prereal_1)
∀x, SNo x(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀qSNoS_ ω, (∀kω, abs_SNo (q + - - x) < eps_ k)q = - x)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_prereal_2)
∀x, SNo x(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)(∀kω, ∃qSNoS_ ω, q < - x - x < q + eps_ k)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_prereal_incr_lower_pos)
∀x, SNo x0 < x(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)∀kω, ∀p : prop, (∀qSNoS_ ω, 0 < qq < xx < q + eps_ kp)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_prereal_incr_lower_approx)
∀x, SNo x(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)∃fSNoS_ ωω, ∀nω, f n < x x < f n + eps_ n ∀in, f i < f n
Proof:
The rest of this subproof is missing.
Theorem. (SNo_prereal_decr_upper_approx)
∀x, SNo x(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)∃gSNoS_ ωω, ∀nω, g n + - eps_ n < x x < g n ∀in, g n < g i
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_lim)
∀lambda, ordinal lambda(∀alphalambda, ordsucc alpha lambda)∀L RSNoS_ lambda, SNoCutP L RSNoLev (SNoCut L R) ordsucc lambda
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_approx_real_lem)
∀f gSNoS_ ωω, (∀n mω, f n < g m)∀p : prop, (SNoCutP {f n|nω} {g n|nω}SNo (SNoCut {f n|nω} {g n|nω})SNoLev (SNoCut {f n|nω} {g n|nω}) ordsucc ωSNoCut {f n|nω} {g n|nω} SNoS_ (ordsucc ω)(∀nω, f n < SNoCut {f n|nω} {g n|nω})(∀nω, SNoCut {f n|nω} {g n|nω} < g n)p)p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_approx_real)
∀x, SNo x∀f gSNoS_ ωω, (∀nω, f n < x)(∀nω, x < f n + eps_ n)(∀nω, ∀in, f i < f n)(∀nω, x < g n)(∀nω, ∀in, g n < g i)x = SNoCut {f n|nω} {g n|nω}x real
Proof:
The rest of this subproof is missing.
Theorem. (SNo_approx_real_rep)
∀xreal, ∀p : prop, (∀f gSNoS_ ωω, (∀nω, f n < x)(∀nω, x < f n + eps_ n)(∀nω, ∀in, f i < f n)(∀nω, g n + - eps_ n < x)(∀nω, x < g n)(∀nω, ∀in, g n < g i)SNoCutP {f n|nω} {g n|nω}x = SNoCut {f n|nω} {g n|nω}p)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (real_mul_SNo_pos)
∀x yreal, 0 < x0 < yx * y real
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (pos_real_left_approx_double)
∀xreal, 0 < xx 2(∀mω, x eps_ m)∃wSNoL_pos x, x < 2 * w
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section Reals
Beginning of Section even_odd
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (nat_le2_cases)
∀m, nat_p mm 2m = 0 m = 1 m = 2
Proof:
The rest of this subproof is missing.
Theorem. (prime_nat_2_lem)
∀m, nat_p m∀n, nat_p nm * n = 2m = 1 m = 2
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Proof:
The rest of this subproof is missing.
End of Section even_odd
Beginning of Section form100_22b
Let tag : setsetλalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Proof:
The rest of this subproof is missing.
Theorem. (Repl_finite)
∀f : setset, ∀X, finite Xfinite {f x|xX}
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section form100_22b
Beginning of Section rational
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Definition. We define rational to be {xreal|∃mint, ∃nω {0}, x = m :/: n} of type set.
End of Section rational
Beginning of Section form100_3
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define nat_pair to be λm n ⇒ 2 ^ m * (2 * n + 1) of type setsetset.
Proof:
The rest of this subproof is missing.
Theorem. (nat_pair_0)
∀m n m' n'ω, nat_pair m n = nat_pair m' n'm = m'
Proof:
The rest of this subproof is missing.
Theorem. (nat_pair_1)
∀m n m' n'ω, nat_pair m n = nat_pair m' n'n = n'
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section form100_3
Beginning of Section SurrealSqrt
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Definition. We define SNoL_nonneg to be λx ⇒ {wSNoL x|0 w} of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define SNo_sqrtauxset to be λY Z x ⇒ yY{(x + y * z) :/: (y + z)|zZ, 0 < y + z} of type setsetsetset.
Theorem. (SNo_sqrtauxset_I)
∀Y Z x, ∀yY, ∀zZ, 0 < y + z(x + y * z) :/: (y + z) SNo_sqrtauxset Y Z x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtauxset_E)
∀Y Z x, ∀uSNo_sqrtauxset Y Z x, ∀p : prop, (∀yY, ∀zZ, 0 < y + zu = (x + y * z) :/: (y + z)p)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define SNo_sqrtaux to be λx g ⇒ nat_primrec ({g w|wSNoL_nonneg x},{g z|zSNoR x}) (λk p ⇒ (p 0 SNo_sqrtauxset (p 0) (p 1) x,p 1 SNo_sqrtauxset (p 0) (p 0) x SNo_sqrtauxset (p 1) (p 1) x)) of type set(setset)setset.
Theorem. (SNo_sqrtaux_0)
∀x, ∀g : setset, SNo_sqrtaux x g 0 = ({g w|wSNoL_nonneg x},{g z|zSNoR x})
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_S)
∀x, ∀g : setset, ∀n, nat_p nSNo_sqrtaux x g (ordsucc n) = (SNo_sqrtaux x g n 0 SNo_sqrtauxset (SNo_sqrtaux x g n 0) (SNo_sqrtaux x g n 1) x,SNo_sqrtaux x g n 1 SNo_sqrtauxset (SNo_sqrtaux x g n 0) (SNo_sqrtaux x g n 0) x SNo_sqrtauxset (SNo_sqrtaux x g n 1) (SNo_sqrtaux x g n 1) x)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_mon_lem)
∀x, ∀g : setset, ∀m, nat_p m∀n, nat_p nSNo_sqrtaux x g m 0 SNo_sqrtaux x g (add_nat m n) 0 SNo_sqrtaux x g m 1 SNo_sqrtaux x g (add_nat m n) 1
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_mon)
∀x, ∀g : setset, ∀m, nat_p m∀n, nat_p nm nSNo_sqrtaux x g m 0 SNo_sqrtaux x g n 0 SNo_sqrtaux x g m 1 SNo_sqrtaux x g n 1
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_ext)
∀x, SNo x∀g h : setset, (∀x'SNoS_ (SNoLev x), g x' = h x')∀k, nat_p kSNo_sqrtaux x g k = SNo_sqrtaux x h k
Proof:
The rest of this subproof is missing.
Beginning of Section sqrt_SNo_nonneg
Let G : set(setset)setλx g ⇒ SNoCut (kωSNo_sqrtaux x g k 0) (kωSNo_sqrtaux x g k 1)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section sqrt_SNo_nonneg
Theorem. (SNo_sqrtaux_0_1_prop)
∀x, SNo x0 x∀k, nat_p k(∀ySNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y 0 y y * y < x) (∀ySNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y 0 y x < y * y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_0_prop)
∀x, SNo x0 x∀k, nat_p k∀ySNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y 0 y y * y < x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_1_prop)
∀x, SNo x0 x∀k, nat_p k∀ySNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y 0 y x < y * y
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SurrealSqrt
Beginning of Section form100_1
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Proof:
The rest of this subproof is missing.
Theorem. (form100_1_lem1)
∀m, nat_p m∀n, nat_p nm * m = 2 * n * nn = 0
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Proof:
The rest of this subproof is missing.
End of Section form100_1
Beginning of Section Topology
Definition. We define topology_on to be λX T ⇒ T 𝒫 X Empty T X T (∀UFam𝒫 T, UFam T) (∀UT, ∀VT, U V T) of type setsetprop.
Definition. We define open_in to be λX T U ⇒ topology_on X T U T of type setsetsetprop.
Axiom. (open_in_subset) We take the following as an axiom:
∀X T U : set, open_in X T UU X
Axiom. (topology_elem_subset) We take the following as an axiom:
∀X T U : set, topology_on X TU TU X
Axiom. (topology_has_empty) We take the following as an axiom:
∀X T : set, topology_on X TEmpty T
Axiom. (topology_has_X) We take the following as an axiom:
∀X T : set, topology_on X TX T
Axiom. (topology_union_closed) We take the following as an axiom:
∀X T UFam : set, topology_on X TUFam T UFam T
Axiom. (topology_binintersect_closed) We take the following as an axiom:
∀X T U V : set, topology_on X TU TV TU V T
Definition. We define closed_in to be λX T C ⇒ topology_on X T (C X ∃UT, C = X U) of type setsetsetprop.
Theorem. (closed_of_open_complement)
∀X T U : set, topology_on X TU Tclosed_in X T (X U)
Proof:
Let X be given.
Let T be given.
Let U be given.
Assume Htop HU.
We will prove topology_on X T (X U X ∃VT, X U = X V).
Apply andI to the current goal.
An exact proof term for the current goal is Htop.
Apply andI to the current goal.
Apply setminus_Subq to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HU.
Use reflexivity.
Definition. We define finer_than to be λT' T ⇒ T T' of type setsetprop.
Definition. We define coarser_than to be λT' T ⇒ T' T of type setsetprop.
Definition. We define discrete_topology to be λX ⇒ 𝒫 X of type setset.
Definition. We define indiscrete_topology to be λX ⇒ {Empty,X} of type setset.
Definition. We define finite_complement_topology to be λX ⇒ {U𝒫 X|finite (X U) U = Empty} of type setset.
Definition. We define countable to be λX ⇒ atleastp X ω of type setprop.
Proof:
Let X be given.
Assume Hfin.
Apply Hfin to the current goal.
Let n be given.
Assume Hpair: n ω equip X n.
We prove the intermediate claim Hn: n ω.
An exact proof term for the current goal is (andEL (n ω) (equip X n) Hpair).
We prove the intermediate claim Heq: equip X n.
An exact proof term for the current goal is (andER (n ω) (equip X n) Hpair).
We prove the intermediate claim Hn_sub: n ω.
An exact proof term for the current goal is (omega_TransSet n Hn).
We prove the intermediate claim Hcount_n: atleastp n ω.
An exact proof term for the current goal is (Subq_atleastp n ω Hn_sub).
We prove the intermediate claim Hcount_X: atleastp X n.
An exact proof term for the current goal is (equip_atleastp X n Heq).
An exact proof term for the current goal is (atleastp_tra X n ω Hcount_X Hcount_n).
Proof:
An exact proof term for the current goal is (Subq_atleastp Empty ω (Subq_Empty ω)).
Theorem. (Subq_countable)
∀X Y : set, countable YX Ycountable X
Proof:
Let X and Y be given.
Assume HcountY HsubXY.
We will prove atleastp X ω.
Apply atleastp_tra X Y ω to the current goal.
An exact proof term for the current goal is (Subq_atleastp X Y HsubXY).
An exact proof term for the current goal is HcountY.
Axiom. (binunion_countable) We take the following as an axiom:
∀X Y : set, countable Xcountable Ycountable (X Y)
Axiom. (Union_Power) We take the following as an axiom:
∀X Fam : set, Fam 𝒫 X Fam X
Axiom. (binintersect_Power) We take the following as an axiom:
∀X U V : set, U 𝒫 XV 𝒫 XU V 𝒫 X
Axiom. (setminus_Power) We take the following as an axiom:
∀X U V : set, U 𝒫 XX U 𝒫 X
Definition. We define countable_complement_topology to be λX ⇒ {U𝒫 X|countable (X U) U = Empty} of type setset.
Proof:
Let X be given.
We will prove 𝒫 X 𝒫 X Empty 𝒫 X X 𝒫 X (∀UFam𝒫 (𝒫 X), UFam 𝒫 X) (∀U𝒫 X, ∀V𝒫 X, U V 𝒫 X).
Apply andI to the current goal.
We will prove ((𝒫 X 𝒫 X Empty 𝒫 X) X 𝒫 X (∀UFam𝒫 (𝒫 X), UFam 𝒫 X)).
Apply andI to the current goal.
We will prove 𝒫 X 𝒫 X Empty 𝒫 X X 𝒫 X.
Apply andI to the current goal.
We will prove 𝒫 X 𝒫 X Empty 𝒫 X.
Apply andI to the current goal.
An exact proof term for the current goal is (Subq_ref (𝒫 X)).
Apply Empty_In_Power to the current goal.
Apply PowerI to the current goal.
An exact proof term for the current goal is (Subq_ref X).
We will prove ∀UFam𝒫 (𝒫 X), UFam 𝒫 X.
Let UFam be given.
Assume Hfam: UFam 𝒫 (𝒫 X).
Apply PowerI X ( UFam) to the current goal.
Let x be given.
Assume HxUnion: x UFam.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HUinx: x U.
Assume HUinFam: U UFam.
We prove the intermediate claim HFamSub: UFam 𝒫 X.
An exact proof term for the current goal is (iffEL (UFam 𝒫 (𝒫 X)) (UFam 𝒫 X) (PowerEq (𝒫 X) UFam) Hfam).
We prove the intermediate claim HUinPower: U 𝒫 X.
An exact proof term for the current goal is HFamSub U HUinFam.
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (iffEL (U 𝒫 X) (U X) (PowerEq X U) HUinPower).
An exact proof term for the current goal is (HUsub x HUinx).
We will prove ∀U𝒫 X, ∀V𝒫 X, U V 𝒫 X.
Let U be given.
Assume HU: U 𝒫 X.
Let V be given.
Assume HV: V 𝒫 X.
Apply PowerI X (U V) to the current goal.
Let x be given.
Assume Hxcap: x U V.
Apply binintersectE U V x Hxcap to the current goal.
Assume HxU HxV.
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (iffEL (U 𝒫 X) (U X) (PowerEq X U) HU).
An exact proof term for the current goal is (HUsub x HxU).
Proof:
Let X be given.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Let U be given.
Assume HU: U indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUe: U = Empty.
rewrite the current goal using HUe (from left to right).
An exact proof term for the current goal is (Empty_In_Power X).
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (Self_In_Power X).
An exact proof term for the current goal is (UPairI1 Empty X).
An exact proof term for the current goal is (UPairI2 Empty X).
We will prove ∀UFam𝒫 (indiscrete_topology X), UFam indiscrete_topology X.
Let UFam be given.
Assume Hfam: UFam 𝒫 (indiscrete_topology X).
We prove the intermediate claim Hsub: UFam indiscrete_topology X.
An exact proof term for the current goal is (PowerE (indiscrete_topology X) UFam Hfam).
Apply xm (∃U : set, U UFam U = X) to the current goal.
Assume Hex: ∃U : set, U UFam U = X.
We prove the intermediate claim HUnion_sub: UFam X.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U indiscrete_topology X.
An exact proof term for the current goal is (Hsub U HUin).
Apply UPairE U Empty X HUtop to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim HxEmpty: x Empty.
rewrite the current goal using HUe (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (EmptyE x HxEmpty (x X)).
Assume HUX: U = X.
rewrite the current goal using HUX (from right to left).
An exact proof term for the current goal is HxU.
We prove the intermediate claim HX_sub: X UFam.
Let x be given.
Assume HxX.
Apply Hex to the current goal.
Let U be given.
Assume HUinpair: U UFam U = X.
We prove the intermediate claim HUin: U UFam.
An exact proof term for the current goal is (andEL (U UFam) (U = X) HUinpair).
We prove the intermediate claim HUeq: U = X.
An exact proof term for the current goal is (andER (U UFam) (U = X) HUinpair).
We prove the intermediate claim HxU: x U.
rewrite the current goal using HUeq (from left to right).
An exact proof term for the current goal is HxX.
Apply UnionI UFam x U HxU HUin to the current goal.
We prove the intermediate claim HUnion_eq: UFam = X.
Apply set_ext to the current goal.
An exact proof term for the current goal is HUnion_sub.
An exact proof term for the current goal is HX_sub.
rewrite the current goal using HUnion_eq (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
Assume Hnone: ¬ ∃U : set, U UFam U = X.
We prove the intermediate claim HUnion_empty: UFam = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U indiscrete_topology X.
An exact proof term for the current goal is (Hsub U HUin).
Apply UPairE U Empty X HUtop to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim HxEmpty: x Empty.
rewrite the current goal using HUe (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is HxEmpty.
Assume HUX: U = X.
Apply FalseE to the current goal.
Apply Hnone to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUin.
An exact proof term for the current goal is HUX.
rewrite the current goal using HUnion_empty (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
Let U be given.
Assume HU: U indiscrete_topology X.
Let V be given.
Assume HV: V indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim Hcap: U V = Empty.
rewrite the current goal using HUe (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_1 Empty V).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
Assume HUX: U = X.
Apply UPairE V Empty X HV to the current goal.
Assume HVe: V = Empty.
We prove the intermediate claim Hcap: U V = Empty.
rewrite the current goal using HVe (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 U Empty).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
Assume HVX: V = X.
We prove the intermediate claim Hcap: U V = X.
Apply set_ext to the current goal.
rewrite the current goal using HUX (from left to right).
rewrite the current goal using HVX (from left to right).
An exact proof term for the current goal is (binintersect_Subq_1 X X).
Let x be given.
Assume HxX.
rewrite the current goal using HUX (from left to right).
rewrite the current goal using HVX (from left to right).
An exact proof term for the current goal is (binintersectI X X x HxX HxX).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
Proof:
Let X be given.
We prove the intermediate claim HEmptyOpen: Empty finite_complement_topology X.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X Empty)) (Empty = Empty) (λP H ⇒ H))).
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Let U be given.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X Empty)) (Empty = Empty) (λP H ⇒ H))).
We prove the intermediate claim Hdiff_empty: X X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot HxX).
We prove the intermediate claim HfinDiff: finite (X X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is finite_Empty.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) X (Self_In_Power X) (orIL (finite (X X)) (X = Empty) HfinDiff)).
Let UFam be given.
Assume Hfam: UFam 𝒫 (finite_complement_topology X).
We prove the intermediate claim Hsub: UFam finite_complement_topology X.
An exact proof term for the current goal is (PowerE (finite_complement_topology X) UFam Hfam).
Apply xm (∃U : set, U UFam finite (X U)) to the current goal.
Assume Hex: ∃U : set, U UFam finite (X U).
We prove the intermediate claim HUnionInPower: UFam 𝒫 X.
Apply PowerI X ( UFam) to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUinPow: U 𝒫 X.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U (Hsub U HUin)).
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (PowerE X U HUinPow).
An exact proof term for the current goal is (HUsub x HxU).
We prove the intermediate claim HUnionPred: finite (X UFam) UFam = Empty.
Apply orIL to the current goal.
Apply Hex to the current goal.
Let U be given.
Assume Hpair: U UFam finite (X U).
We prove the intermediate claim HUin: U UFam.
An exact proof term for the current goal is (andEL (U UFam) (finite (X U)) Hpair).
We prove the intermediate claim HUfin: finite (X U).
An exact proof term for the current goal is (andER (U UFam) (finite (X U)) Hpair).
We prove the intermediate claim Hsubset: X UFam X U.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X ( UFam) x Hx).
We prove the intermediate claim HnotUnion: x UFam.
An exact proof term for the current goal is (setminusE2 X ( UFam) x Hx).
We prove the intermediate claim HnotU: x U.
Assume HxU.
Apply HnotUnion to the current goal.
Apply UnionI UFam x U HxU HUin to the current goal.
Apply setminusI X U x HxX HnotU to the current goal.
An exact proof term for the current goal is (Subq_finite (X U) HUfin (X UFam) Hsubset).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) ( UFam) HUnionInPower HUnionPred).
Assume Hnone: ¬ ∃U : set, U UFam finite (X U).
We prove the intermediate claim HUnionEmpty: UFam = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUdata: finite (X U) U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U (Hsub U HUin)).
Apply HUdata (x Empty) to the current goal.
Assume HUfin.
Apply FalseE to the current goal.
Apply Hnone to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUin.
An exact proof term for the current goal is HUfin.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from right to left).
An exact proof term for the current goal is HxU.
rewrite the current goal using HUnionEmpty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
Let U be given.
Let V be given.
We prove the intermediate claim HUdata: finite (X U) U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
We prove the intermediate claim HVdata: finite (X V) V = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) V HV).
Apply HUdata (U V finite_complement_topology X) to the current goal.
Assume HUfin.
Apply HVdata (U V finite_complement_topology X) to the current goal.
Assume HVfin.
We prove the intermediate claim HcapInPower: U V 𝒫 X.
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU)).
Apply PowerI X (U V) to the current goal.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
An exact proof term for the current goal is (HUsub x HxU).
We prove the intermediate claim HcapPred: finite (X (U V)) U V = Empty.
Apply orIL to the current goal.
We prove the intermediate claim HfinUnion: finite ((X U) (X V)).
An exact proof term for the current goal is (binunion_finite (X U) HUfin (X V) HVfin).
We prove the intermediate claim Hsubset: X (U V) (X U) (X V).
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X (U V) x Hx).
We prove the intermediate claim HnotCap: x U V.
An exact proof term for the current goal is (setminusE2 X (U V) x Hx).
Apply xm (x U) to the current goal.
Assume HxU.
We prove the intermediate claim HnotV: x V.
Assume HxV.
Apply HnotCap to the current goal.
An exact proof term for the current goal is (binintersectI U V x HxU HxV).
Apply binunionI2 (X U) (X V) to the current goal.
Apply setminusI X V x HxX HnotV to the current goal.
Assume HnotU.
Apply binunionI1 (X U) (X V) to the current goal.
Apply setminusI X U x HxX HnotU to the current goal.
An exact proof term for the current goal is (Subq_finite ((X U) (X V)) HfinUnion (X (U V)) Hsubset).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) (U V) HcapInPower HcapPred).
Assume HVempty: V = Empty.
We prove the intermediate claim Hcap_empty: U V = Empty.
rewrite the current goal using HVempty (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 U Empty).
rewrite the current goal using Hcap_empty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
Assume HUempty: U = Empty.
We prove the intermediate claim Hcap_empty: U V = Empty.
rewrite the current goal using HUempty (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_1 Empty V).
rewrite the current goal using Hcap_empty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
Proof:
Let T be given.
An exact proof term for the current goal is (Subq_ref T).
Theorem. (finer_than_trans)
∀A B C : set, finer_than B Afiner_than C Bfiner_than C A
Proof:
Let A, B and C be given.
Assume H1: finer_than B A.
Assume H2: finer_than C B.
An exact proof term for the current goal is (Subq_tra A B C H1 H2).
Proof:
Let T and T' be given.
Assume H.
An exact proof term for the current goal is H.
Definition. We define comparable_topologies to be λT1 T2 ⇒ finer_than T1 T2 finer_than T2 T1 of type setsetprop.
Definition. We define topology_eq to be λX T1 T2 ⇒ topology_on X T1 topology_on X T2 T1 = T2 of type setsetsetprop.
Theorem. (topology_eq_sym)
∀X T1 T2 : set, topology_eq X T1 T2topology_eq X T2 T1
Proof:
Let X, T1 and T2 be given.
Assume H.
We prove the intermediate claim Hpair: topology_on X T1 topology_on X T2.
An exact proof term for the current goal is (andEL (topology_on X T1 topology_on X T2) (T1 = T2) H).
We prove the intermediate claim Heq: T1 = T2.
An exact proof term for the current goal is (andER (topology_on X T1 topology_on X T2) (T1 = T2) H).
We prove the intermediate claim HT1: topology_on X T1.
An exact proof term for the current goal is (andEL (topology_on X T1) (topology_on X T2) Hpair).
We prove the intermediate claim HT2: topology_on X T2.
An exact proof term for the current goal is (andER (topology_on X T1) (topology_on X T2) Hpair).
We will prove topology_on X T2 topology_on X T1 T2 = T1.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT2.
An exact proof term for the current goal is HT1.
rewrite the current goal using Heq (from right to left).
Use reflexivity.
Theorem. (topology_eq_trans)
∀X T1 T2 T3 : set, topology_eq X T1 T2topology_eq X T2 T3topology_eq X T1 T3
Proof:
Let X, T1, T2 and T3 be given.
Assume H12 H23.
We prove the intermediate claim H12pair: topology_on X T1 topology_on X T2.
An exact proof term for the current goal is (andEL (topology_on X T1 topology_on X T2) (T1 = T2) H12).
We prove the intermediate claim H12eq: T1 = T2.
An exact proof term for the current goal is (andER (topology_on X T1 topology_on X T2) (T1 = T2) H12).
We prove the intermediate claim HT1: topology_on X T1.
An exact proof term for the current goal is (andEL (topology_on X T1) (topology_on X T2) H12pair).
We prove the intermediate claim HT2: topology_on X T2.
An exact proof term for the current goal is (andER (topology_on X T1) (topology_on X T2) H12pair).
We prove the intermediate claim H23pair: topology_on X T2 topology_on X T3.
An exact proof term for the current goal is (andEL (topology_on X T2 topology_on X T3) (T2 = T3) H23).
We prove the intermediate claim H23eq: T2 = T3.
An exact proof term for the current goal is (andER (topology_on X T2 topology_on X T3) (T2 = T3) H23).
We prove the intermediate claim HT3: topology_on X T3.
An exact proof term for the current goal is (andER (topology_on X T2) (topology_on X T3) H23pair).
We will prove topology_on X T1 topology_on X T3 T1 = T3.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT1.
An exact proof term for the current goal is HT3.
rewrite the current goal using H12eq (from left to right).
rewrite the current goal using H23eq (from left to right).
Use reflexivity.
Proof:
Let X and T be given.
Assume HT.
We will prove topology_on X T topology_on X T T = T.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT.
An exact proof term for the current goal is HT.
Use reflexivity.
Definition. We define strictly_finer_than to be λT' T ⇒ finer_than T' T ¬ finer_than T T' of type setsetprop.
Definition. We define strictly_coarser_than to be λT' T ⇒ coarser_than T' T ¬ coarser_than T T' of type setsetprop.
Definition. We define discrete_topology_alt to be discrete_topology of type setset.
Definition. We define trivial_topology to be indiscrete_topology of type setset.
Definition. We define finer_than_topology to be λX T' T ⇒ topology_on X T' topology_on X T finer_than T' T of type setsetsetprop.
Proof:
Let T and T' be given.
Apply iffI to the current goal.
Assume H.
An exact proof term for the current goal is H.
Assume H.
An exact proof term for the current goal is H.
Proof:
Let X and T be given.
Assume HT.
We prove the intermediate claim H1: ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T).
An exact proof term for the current goal is (andEL (((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T)) (∀UT, ∀VT, U V T) HT).
We prove the intermediate claim H2: (T 𝒫 X Empty T) X T.
An exact proof term for the current goal is (andEL ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T) H1).
We prove the intermediate claim H3: T 𝒫 X Empty T.
An exact proof term for the current goal is (andEL (T 𝒫 X Empty T) (X T) H2).
We prove the intermediate claim HTsub: T 𝒫 X.
An exact proof term for the current goal is (andEL (T 𝒫 X) (Empty T) H3).
An exact proof term for the current goal is HTsub.
Proof:
Let X and T be given.
Assume HT.
We prove the intermediate claim Hchunk1: ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T).
An exact proof term for the current goal is (andEL (((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T)) (∀UT, ∀VT, U V T) HT).
We prove the intermediate claim Hchunk2: (T 𝒫 X Empty T) X T.
An exact proof term for the current goal is (andEL ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T) Hchunk1).
We prove the intermediate claim Hchunk3: T 𝒫 X Empty T.
An exact proof term for the current goal is (andEL (T 𝒫 X Empty T) (X T) Hchunk2).
We prove the intermediate claim Hempty: Empty T.
An exact proof term for the current goal is (andER (T 𝒫 X) (Empty T) Hchunk3).
We prove the intermediate claim HX: X T.
An exact proof term for the current goal is (andER ((T 𝒫 X) Empty T) (X T) Hchunk2).
Let U be given.
Assume HU: U indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from left to right).
An exact proof term for the current goal is Hempty.
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is HX.
Proof:
Let X and U be given.
Assume HUsub.
Apply PowerI X U HUsub to the current goal.
Proof:
Let X and U be given.
Apply iffI to the current goal.
Assume HU.
An exact proof term for the current goal is (UPairE U Empty X HU).
Assume Hcases: U = Empty U = X.
We prove the intermediate claim HUempty_branch: U = EmptyU indiscrete_topology X.
Assume HUE: U = Empty.
rewrite the current goal using HUE (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
We prove the intermediate claim HUx_branch: U = XU indiscrete_topology X.
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
An exact proof term for the current goal is (Hcases (U indiscrete_topology X) HUempty_branch HUx_branch).
Proof:
Let X and U be given.
Assume Hopen.
We prove the intermediate claim HUin: U finite_complement_topology X.
An exact proof term for the current goal is (andER (topology_on X (finite_complement_topology X)) (U finite_complement_topology X) Hopen).
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HUin).
Proof:
Let X be given.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X Empty)) (Empty = Empty) (λP H ⇒ H))).
Proof:
Let X be given.
We prove the intermediate claim Hdiff_empty: X X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim Hxin: x X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot Hxin).
We prove the intermediate claim HfinDiff: finite (X X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is finite_Empty.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) X (Self_In_Power X) (orIL (finite (X X)) (X = Empty) HfinDiff)).
Proof:
Let X and U be given.
Assume Hopen.
We prove the intermediate claim HUin: U countable_complement_topology X.
An exact proof term for the current goal is (andER (topology_on X (countable_complement_topology X)) (U countable_complement_topology X) Hopen).
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setcountable (X U0) U0 = Empty) U HUin).
Proof:
Let X be given.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setcountable (X U0) U0 = Empty) Empty (Empty_In_Power X) (orIR (countable (X Empty)) (Empty = Empty) (λP H ⇒ H))).
Proof:
Let X be given.
We prove the intermediate claim Hdiff_empty: X X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot HxX).
We prove the intermediate claim HcountDiff: countable (X X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is (Subq_atleastp Empty ω (Subq_Empty ω)).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setcountable (X U0) U0 = Empty) X (Self_In_Power X) (orIL (countable (X X)) (X = Empty) HcountDiff)).
Proof:
Let X be given.
Let U be given.
We prove the intermediate claim HUinPow: U 𝒫 X.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
We prove the intermediate claim HUdata: finite (X U) U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
We prove the intermediate claim HUpred: countable (X U) U = Empty.
Apply HUdata (countable (X U) U = Empty) to the current goal.
Assume HUfin: finite (X U).
Apply orIL to the current goal.
An exact proof term for the current goal is (finite_countable (X U) HUfin).
Assume HUemp: U = Empty.
Apply orIR to the current goal.
An exact proof term for the current goal is HUemp.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setcountable (X U0) U0 = Empty) U HUinPow HUpred).
Proof:
Let X be given.
Let U be given.
Assume HU.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
Proof:
Let X be given.
Let U be given.
Assume HU: U indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from left to right).
An exact proof term for the current goal is (countable_complement_topology_contains_empty X).
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (countable_complement_topology_contains_full X).
Definition. We define finer_than_topology_by_inclusion to be λX T' T ⇒ topology_on X T' topology_on X T T T' of type setsetsetprop.
Proof:
Let X, T' and T be given.
Apply iffI to the current goal.
Assume H.
An exact proof term for the current goal is H.
Assume H.
An exact proof term for the current goal is H.
Theorem. (lemma_union_of_topology_family_open)
∀X T UFam : set, topology_on X TUFam 𝒫 T UFam T
Proof:
Let X, T and UFam be given.
Assume HT Hfam.
We prove the intermediate claim Hchunk1: ((T 𝒫 X Empty T) X T) (∀UFam'𝒫 T, UFam' T).
An exact proof term for the current goal is (andEL (((T 𝒫 X Empty T) X T) (∀UFam'𝒫 T, UFam' T)) (∀UT, ∀VT, U V T) HT).
We prove the intermediate claim Hunion_axiom: ∀UFam'𝒫 T, UFam' T.
An exact proof term for the current goal is (andER ((T 𝒫 X Empty T) X T) (∀UFam'𝒫 T, UFam' T) Hchunk1).
An exact proof term for the current goal is (Hunion_axiom UFam Hfam).
Theorem. (lemma_intersection_two_open)
∀X T U V : set, topology_on X TU TV TU V T
Proof:
Let X, T, U and V be given.
Assume HT HU HV.
We prove the intermediate claim Hax_inter: ∀U'T, ∀V'T, U' V' T.
An exact proof term for the current goal is (andER ((T 𝒫 X Empty T X T (∀UFam𝒫 T, UFam T))) (∀U'T, ∀V'T, U' V' T) HT).
An exact proof term for the current goal is (Hax_inter U HU V HV).
Definition. We define topological_space to be topology_on of type setsetprop.
Definition. We define open_set_family to be λ_ T ⇒ T of type setsetset.
Definition. We define open_set to be λX T U ⇒ topology_on X T U T of type setsetsetprop.
Definition. We define basis_on to be λX B ⇒ B 𝒫 X (∀xX, ∃bB, x b) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) of type setsetprop.
Definition. We define generated_topology to be λX B ⇒ {U𝒫 X|∀xU, ∃bB, x b b U} of type setsetset.
Axiom. (basis_elem_subset) We take the following as an axiom:
∀X B b : set, basis_on X Bb Bb X
Axiom. (basis_in_generated) We take the following as an axiom:
∀X B b : set, basis_on X Bb Bb generated_topology X B
Axiom. (generated_topology_subset) We take the following as an axiom:
∀X B U : set, U generated_topology X BU X
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBleft: B 𝒫 X (∀xX, ∃bB, x b).
An exact proof term for the current goal is (andEL (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis).
We prove the intermediate claim HBint: ∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2.
An exact proof term for the current goal is (andER (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis).
We prove the intermediate claim HBsub: B 𝒫 X.
An exact proof term for the current goal is (andEL (B 𝒫 X) (∀xX, ∃bB, x b) HBleft).
We prove the intermediate claim HBcov: ∀xX, ∃bB, x b.
An exact proof term for the current goal is (andER (B 𝒫 X) (∀xX, ∃bB, x b) HBleft).
We prove the intermediate claim proofA: generated_topology X B 𝒫 X.
Let U be given.
Assume HU: U generated_topology X B.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) U HU).
We prove the intermediate claim proofB: Empty generated_topology X B.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) Empty (Empty_In_Power X) (λx HxEmpty ⇒ EmptyE x HxEmpty (∃bB, x b b Empty))).
We prove the intermediate claim proofC: X generated_topology X B.
We prove the intermediate claim HXprop: ∀xX, ∃bB, x b b X.
Let x be given.
Assume HxX.
We prove the intermediate claim Hexb: ∃bB, x b.
An exact proof term for the current goal is (HBcov x HxX).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andER (b B) (x b) Hbpair).
We prove the intermediate claim HbsubX: b X.
An exact proof term for the current goal is (PowerE X b (HBsub b HbB)).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
An exact proof term for the current goal is HbsubX.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) X (Self_In_Power X) HXprop).
We prove the intermediate claim proofD: ∀UFam𝒫 (generated_topology X B), UFam generated_topology X B.
Let UFam be given.
Assume Hfam: UFam 𝒫 (generated_topology X B).
We prove the intermediate claim HsubFam: UFam generated_topology X B.
An exact proof term for the current goal is (PowerE (generated_topology X B) UFam Hfam).
We prove the intermediate claim HPowUnion: UFam 𝒫 X.
Apply PowerI X ( UFam) to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U generated_topology X B.
An exact proof term for the current goal is (HsubFam U HUin).
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop)).
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HUnionProp: ∀x UFam, ∃bB, x b b UFam.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U generated_topology X B.
An exact proof term for the current goal is (HsubFam U HUin).
We prove the intermediate claim HUprop: ∀x0U, ∃bB, x0 b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop).
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbSubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
Let y be given.
Assume Hyb.
Apply UnionI UFam y U (HbSubU y Hyb) HUin to the current goal.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) ( UFam) HPowUnion HUnionProp).
We prove the intermediate claim proofE: ∀Ugenerated_topology X B, ∀Vgenerated_topology X B, U V generated_topology X B.
Let U be given.
Assume HUtop.
Let V be given.
Assume HVtop.
We prove the intermediate claim HUprop: ∀x0U, ∃bB, x0 b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop).
We prove the intermediate claim HVprop: ∀x0V, ∃bB, x0 b b V.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) V HVtop).
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop)).
We prove the intermediate claim HPowCap: U V 𝒫 X.
Apply PowerI X (U V) to the current goal.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HCapProp: ∀xU V, ∃bB, x b b U V.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
We prove the intermediate claim Hexb1: ∃b1B, x b1 b1 U.
An exact proof term for the current goal is (HUprop x HxU).
We prove the intermediate claim Hexb2: ∃b2B, x b2 b2 V.
An exact proof term for the current goal is (HVprop x HxV).
Apply Hexb1 to the current goal.
Let b1 be given.
Assume Hbpair1.
We prove the intermediate claim Hb1: b1 B.
An exact proof term for the current goal is (andEL (b1 B) (x b1 b1 U) Hbpair1).
We prove the intermediate claim Hb1prop: x b1 b1 U.
An exact proof term for the current goal is (andER (b1 B) (x b1 b1 U) Hbpair1).
We prove the intermediate claim Hb1x: x b1.
An exact proof term for the current goal is (andEL (x b1) (b1 U) Hb1prop).
We prove the intermediate claim Hb1Sub: b1 U.
An exact proof term for the current goal is (andER (x b1) (b1 U) Hb1prop).
Apply Hexb2 to the current goal.
Let b2 be given.
Assume Hbpair2.
We prove the intermediate claim Hb2: b2 B.
An exact proof term for the current goal is (andEL (b2 B) (x b2 b2 V) Hbpair2).
We prove the intermediate claim Hb2prop: x b2 b2 V.
An exact proof term for the current goal is (andER (b2 B) (x b2 b2 V) Hbpair2).
We prove the intermediate claim Hb2x: x b2.
An exact proof term for the current goal is (andEL (x b2) (b2 V) Hb2prop).
We prove the intermediate claim Hb2Sub: b2 V.
An exact proof term for the current goal is (andER (x b2) (b2 V) Hb2prop).
We prove the intermediate claim Hexb3: ∃b3B, x b3 b3 b1 b2.
An exact proof term for the current goal is (HBint b1 Hb1 b2 Hb2 x Hb1x Hb2x).
Apply Hexb3 to the current goal.
Let b3 be given.
Assume Hbpair3.
We prove the intermediate claim Hb3: b3 B.
An exact proof term for the current goal is (andEL (b3 B) (x b3 b3 b1 b2) Hbpair3).
We prove the intermediate claim Hb3prop: x b3 b3 b1 b2.
An exact proof term for the current goal is (andER (b3 B) (x b3 b3 b1 b2) Hbpair3).
We prove the intermediate claim HxB3: x b3.
An exact proof term for the current goal is (andEL (x b3) (b3 b1 b2) Hb3prop).
We prove the intermediate claim Hb3Sub: b3 b1 b2.
An exact proof term for the current goal is (andER (x b3) (b3 b1 b2) Hb3prop).
We use b3 to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb3.
Apply andI to the current goal.
An exact proof term for the current goal is HxB3.
Let y be given.
Assume Hyb3.
We prove the intermediate claim Hy_in_b1b2: y b1 b2.
An exact proof term for the current goal is (Hb3Sub y Hyb3).
Apply binintersectE b1 b2 y Hy_in_b1b2 to the current goal.
Assume Hyb1 Hyb2.
Apply binintersectI U V y (Hb1Sub y Hyb1) (Hb2Sub y Hyb2) to the current goal.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) (U V) HPowCap HCapProp).
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is proofA.
An exact proof term for the current goal is proofB.
An exact proof term for the current goal is proofC.
An exact proof term for the current goal is proofD.
An exact proof term for the current goal is proofE.
Theorem. (generated_topology_contains_basis)
∀X B : set, basis_on X B∀b : set, b Bb generated_topology X B
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B 𝒫 X.
An exact proof term for the current goal is (andEL (B 𝒫 X) (∀xX, ∃bB, x b) (andEL (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis)).
We prove the intermediate claim HBint: ∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2.
An exact proof term for the current goal is (andER (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis).
Let b0 be given.
Assume Hb0.
We prove the intermediate claim Hb0_subX: b0 X.
An exact proof term for the current goal is (PowerE X b0 (HBsub b0 Hb0)).
We prove the intermediate claim Hb0prop: ∀xb0, ∃bB, x b b b0.
Let x be given.
Assume Hxb0.
We prove the intermediate claim Hexb3: ∃b3B, x b3 b3 b0 b0.
An exact proof term for the current goal is (HBint b0 Hb0 b0 Hb0 x Hxb0 Hxb0).
Apply Hexb3 to the current goal.
Let b3 be given.
Assume Hb3pair.
We prove the intermediate claim Hb3: b3 B.
An exact proof term for the current goal is (andEL (b3 B) (x b3 b3 b0 b0) Hb3pair).
We prove the intermediate claim Hb3prop: x b3 b3 b0 b0.
An exact proof term for the current goal is (andER (b3 B) (x b3 b3 b0 b0) Hb3pair).
We prove the intermediate claim Hxb3: x b3.
An exact proof term for the current goal is (andEL (x b3) (b3 b0 b0) Hb3prop).
We prove the intermediate claim Hb3sub_inter: b3 b0 b0.
An exact proof term for the current goal is (andER (x b3) (b3 b0 b0) Hb3prop).
We prove the intermediate claim Hb3subb0: b3 b0.
Let y be given.
Assume Hyb3.
We prove the intermediate claim Hycap: y b0 b0.
An exact proof term for the current goal is (Hb3sub_inter y Hyb3).
Apply binintersectE b0 b0 y Hycap to the current goal.
Assume Hy1 Hy2.
An exact proof term for the current goal is Hy1.
We use b3 to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb3.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb3.
An exact proof term for the current goal is Hb3subb0.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) b0 (PowerI X b0 Hb0_subX) Hb0prop).
Definition. We define basis_generates to be λX B T ⇒ basis_on X B generated_topology X B = T of type setsetsetprop.
Definition. We define basis_refines to be λX B T ⇒ topology_on X T (∀UT, ∀xU, ∃bB, x b b U) of type setsetsetprop.
Proof:
Let X and B be given.
Assume HBasis.
Use reflexivity.
Theorem. (open_sets_as_unions_of_basis)
∀X B : set, basis_on X B∀U : set, open_in X (generated_topology X B) U∃Fam𝒫 B, Fam = U
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B 𝒫 X.
An exact proof term for the current goal is (andEL (B 𝒫 X) (∀xX, ∃bB, x b) (andEL (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis)).
Let U be given.
Assume HUopen.
We prove the intermediate claim HUtop: U generated_topology X B.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U generated_topology X B) HUopen).
We prove the intermediate claim HUprop: ∀xU, ∃bB, x b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop).
Set Fam to be the term {bB|b U} of type set.
We prove the intermediate claim HFamPow: Fam 𝒫 B.
Apply PowerI B Fam to the current goal.
Let b be given.
Assume HbFam.
An exact proof term for the current goal is (SepE1 B (λb0 : setb0 U) b HbFam).
We prove the intermediate claim HUnion_eq: Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (SepE2 B (λb0 : setb0 U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We prove the intermediate claim HbFam: b Fam.
An exact proof term for the current goal is (SepI B (λb0 : setb0 U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
We use Fam to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HFamPow.
An exact proof term for the current goal is HUnion_eq.
Theorem. (basis_generates_open_sets)
∀X B : set, basis_on X B∀U : set, (∃Fam𝒫 B, Fam = U)open_in X (generated_topology X B) U
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B 𝒫 X.
An exact proof term for the current goal is (andEL (B 𝒫 X) (∀xX, ∃bB, x b) (andEL (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis)).
Let U be given.
Assume Hex.
We prove the intermediate claim HUGen: U generated_topology X B.
Apply Hex to the current goal.
Let Fam be given.
Assume HFampair.
We prove the intermediate claim HFamPow: Fam 𝒫 B.
An exact proof term for the current goal is (andEL (Fam 𝒫 B) ( Fam = U) HFampair).
We prove the intermediate claim HUnionEq: Fam = U.
An exact proof term for the current goal is (andER (Fam 𝒫 B) ( Fam = U) HFampair).
We prove the intermediate claim HFamSubB: Fam B.
An exact proof term for the current goal is (PowerE B Fam HFamPow).
We prove the intermediate claim HFamSubX: Fam 𝒫 X.
Let b be given.
Assume HbFam.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (HFamSubB b HbFam).
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HUnionSubX: Fam X.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbSubX: b X.
An exact proof term for the current goal is (PowerE X b (HFamSubX b HbFam)).
An exact proof term for the current goal is (HbSubX x Hxb).
We prove the intermediate claim HUnionSubU: Fam U.
rewrite the current goal using HUnionEq (from left to right).
An exact proof term for the current goal is (Subq_ref U).
We prove the intermediate claim HUsubUnion: U Fam.
rewrite the current goal using HUnionEq (from right to left).
An exact proof term for the current goal is (Subq_ref ( Fam)).
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (Subq_tra U ( Fam) X HUsubUnion HUnionSubX).
We prove the intermediate claim HUpropU: ∀xU, ∃bB, x b b U.
Let x be given.
Assume HxU.
We prove the intermediate claim HxUnion: x Fam.
An exact proof term for the current goal is (HUsubUnion x HxU).
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (HFamSubB b HbFam).
We prove the intermediate claim HbSubUnion: b Fam.
Let y be given.
Assume Hyb.
An exact proof term for the current goal is (UnionI Fam y b Hyb HbFam).
We prove the intermediate claim HbSubU: b U.
An exact proof term for the current goal is (Subq_tra b ( Fam) U HbSubUnion HUnionSubU).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
An exact proof term for the current goal is HbSubU.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U (PowerI X U HUsubX) HUpropU).
An exact proof term for the current goal is (andI (topology_on X (generated_topology X B)) (U generated_topology X B) (lemma_topology_from_basis X B HBasis) HUGen).
Theorem. (open_as_union_of_basis_elements)
∀X B : set, basis_on X B∀U : set, open_in X (generated_topology X B) UU = {bB|b U}
Proof:
Let X and B be given.
Assume HBasis.
Let U be given.
Assume HUopen.
We prove the intermediate claim HUtop: U generated_topology X B.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U generated_topology X B) HUopen).
We prove the intermediate claim HUprop: ∀xU, ∃bB, x b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop).
Set Fam to be the term {bB|b U} of type set.
Apply set_ext to the current goal.
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We prove the intermediate claim HbFam: b Fam.
An exact proof term for the current goal is (SepI B (λb0 : setb0 U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (SepE2 B (λb0 : setb0 U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
Theorem. (basis_refines_topology)
∀X T C : set, topology_on X T(∀cC, c T)(∀UT, ∀xU, ∃CxC, x Cx Cx U)basis_on X C generated_topology X C = T
Proof:
Let X, T and C be given.
Assume Htop HCsub Href.
We prove the intermediate claim Hleft: ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T).
An exact proof term for the current goal is (andEL (((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T)) (∀UT, ∀VT, U V T) Htop).
We prove the intermediate claim Hcore: (T 𝒫 X Empty T) X T.
An exact proof term for the current goal is (andEL ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T) Hleft).
We prove the intermediate claim HTPowEmpty: T 𝒫 X Empty T.
An exact proof term for the current goal is (andEL (T 𝒫 X Empty T) (X T) Hcore).
We prove the intermediate claim HTsubPow: T 𝒫 X.
An exact proof term for the current goal is (andEL (T 𝒫 X) (Empty T) HTPowEmpty).
We prove the intermediate claim HXT: X T.
An exact proof term for the current goal is (andER (T 𝒫 X Empty T) (X T) Hcore).
We prove the intermediate claim HUnionClosed: ∀UFam𝒫 T, UFam T.
An exact proof term for the current goal is (andER ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T) Hleft).
We prove the intermediate claim HInterClosed: ∀UT, ∀VT, U V T.
An exact proof term for the current goal is (andER (((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T)) (∀UT, ∀VT, U V T) Htop).
We prove the intermediate claim HBasis: basis_on X C.
We will prove (C 𝒫 X (∀xX, ∃cC, x c) (∀b1C, ∀b2C, ∀x : set, x b1x b2∃b3C, x b3 b3 b1 b2)).
Apply andI to the current goal.
Apply andI to the current goal.
Let c be given.
Assume HcC.
An exact proof term for the current goal is (HTsubPow c (HCsub c HcC)).
Let x be given.
Assume HxX.
We prove the intermediate claim Hex: ∃cC, x c c X.
An exact proof term for the current goal is (Href X HXT x HxX).
Apply Hex to the current goal.
Let c be given.
Assume Hpair.
We prove the intermediate claim HcC: c C.
An exact proof term for the current goal is (andEL (c C) (x c c X) Hpair).
We prove the intermediate claim Hcprop: x c c X.
An exact proof term for the current goal is (andER (c C) (x c c X) Hpair).
We prove the intermediate claim Hxc: x c.
An exact proof term for the current goal is (andEL (x c) (c X) Hcprop).
We use c to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HcC.
An exact proof term for the current goal is Hxc.
Let c1 be given.
Assume Hc1C.
Let c2 be given.
Assume Hc2C.
Let x be given.
Assume Hxc1 Hxc2.
We prove the intermediate claim Hc1T: c1 T.
An exact proof term for the current goal is (HCsub c1 Hc1C).
We prove the intermediate claim Hc2T: c2 T.
An exact proof term for the current goal is (HCsub c2 Hc2C).
We prove the intermediate claim HcapT: c1 c2 T.
An exact proof term for the current goal is (HInterClosed c1 Hc1T c2 Hc2T).
We prove the intermediate claim HxCap: x c1 c2.
An exact proof term for the current goal is (binintersectI c1 c2 x Hxc1 Hxc2).
We prove the intermediate claim Hex: ∃c3C, x c3 c3 c1 c2.
An exact proof term for the current goal is (Href (c1 c2) HcapT x HxCap).
An exact proof term for the current goal is Hex.
We prove the intermediate claim Hgen_sub_T: generated_topology X C T.
Let U be given.
Assume HUgen: U generated_topology X C.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃b0C, x0 b0 b0 U0) U HUgen)).
We prove the intermediate claim HUprop: ∀xU, ∃cC, x c c U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃b0C, x0 b0 b0 U0) U HUgen).
Set Fam to be the term {cC|c U} of type set.
We prove the intermediate claim HFamSubC: Fam C.
Let c be given.
Assume HcFam.
An exact proof term for the current goal is (SepE1 C (λc0 : setc0 U) c HcFam).
We prove the intermediate claim HFamSubT: Fam T.
Let c be given.
Assume HcFam.
We prove the intermediate claim HcC: c C.
An exact proof term for the current goal is (HFamSubC c HcFam).
An exact proof term for the current goal is (HCsub c HcC).
We prove the intermediate claim HFamPowT: Fam 𝒫 T.
An exact proof term for the current goal is (PowerI T Fam HFamSubT).
We prove the intermediate claim HUnionSubU: Fam U.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let c be given.
Assume Hxc HcFam.
We prove the intermediate claim Hcprop: c U.
An exact proof term for the current goal is (SepE2 C (λc0 : setc0 U) c HcFam).
An exact proof term for the current goal is (Hcprop x Hxc).
We prove the intermediate claim HUsubUnion: U Fam.
Let x be given.
Assume HxU.
We prove the intermediate claim Hex: ∃cC, x c c U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hex to the current goal.
Let c be given.
Assume Hcpair.
We prove the intermediate claim HcC: c C.
An exact proof term for the current goal is (andEL (c C) (x c c U) Hcpair).
We prove the intermediate claim Hcprop: x c c U.
An exact proof term for the current goal is (andER (c C) (x c c U) Hcpair).
We prove the intermediate claim Hxc: x c.
An exact proof term for the current goal is (andEL (x c) (c U) Hcprop).
We prove the intermediate claim HcsubU: c U.
An exact proof term for the current goal is (andER (x c) (c U) Hcprop).
We prove the intermediate claim HcFam: c Fam.
An exact proof term for the current goal is (SepI C (λc0 : setc0 U) c HcC HcsubU).
An exact proof term for the current goal is (UnionI Fam x c Hxc HcFam).
We prove the intermediate claim HUnionEqU: Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
An exact proof term for the current goal is (HUnionSubU x HxUnion).
Let x be given.
Assume HxU.
An exact proof term for the current goal is (HUsubUnion x HxU).
We prove the intermediate claim HUnionInT: Fam T.
An exact proof term for the current goal is (HUnionClosed Fam HFamPowT).
rewrite the current goal using HUnionEqU (from right to left).
An exact proof term for the current goal is HUnionInT.
We prove the intermediate claim HT_sub_gen: T generated_topology X C.
Let U be given.
Assume HU: U T.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (HTsubPow U HU)).
We prove the intermediate claim HUprop: ∀xU, ∃cC, x c c U.
Let x be given.
Assume HxU.
An exact proof term for the current goal is (Href U HU x HxU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃b0C, x0 b0 b0 U0) U (PowerI X U HUsubX) HUprop).
We prove the intermediate claim HEqual: generated_topology X C = T.
Apply set_ext to the current goal.
Let U be given.
Assume HUgen.
An exact proof term for the current goal is (Hgen_sub_T U HUgen).
Let U be given.
Assume HU.
An exact proof term for the current goal is (HT_sub_gen U HU).
Apply andI to the current goal.
An exact proof term for the current goal is HBasis.
An exact proof term for the current goal is HEqual.
Theorem. (lemma13_2_basis_from_open_subcollection)
∀X T C : set, topology_on X T(∀cC, c T)(∀UT, ∀xU, ∃cC, x c c U)basis_on X C generated_topology X C = T
Proof:
Let X, T and C be given.
Assume Htop HCsub Href.
An exact proof term for the current goal is (basis_refines_topology X T C Htop HCsub Href).
Theorem. (finer_via_basis)
∀X B B' : set, basis_on X Bbasis_on X B'(∀xX, ∀b : set, b Bx b∃b'B', x b' b' b)finer_than (generated_topology X B') (generated_topology X B)
Proof:
Let X, B and B' be given.
Assume HB HB' Hcond.
We prove the intermediate claim HT: topology_on X (generated_topology X B).
An exact proof term for the current goal is (lemma_topology_from_basis X B HB).
We prove the intermediate claim HRefProp: ∀Ugenerated_topology X B, ∀xU, ∃b'B', x b' b' U.
Let U be given.
Assume HU: U generated_topology X B.
Let x be given.
Assume HxU.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HU)).
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HUprop: ∀x0U, ∃bB, x0 b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HU).
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We prove the intermediate claim Hexb': ∃b'B', x b' b' b.
An exact proof term for the current goal is (Hcond x HxX b HbB Hxb).
Apply Hexb' to the current goal.
Let b' be given.
Assume Hb'pair.
We prove the intermediate claim Hb'B: b' B'.
An exact proof term for the current goal is (andEL (b' B') (x b' b' b) Hb'pair).
We prove the intermediate claim Hb'prop: x b' b' b.
An exact proof term for the current goal is (andER (b' B') (x b' b' b) Hb'pair).
We prove the intermediate claim Hxb': x b'.
An exact proof term for the current goal is (andEL (x b') (b' b) Hb'prop).
We prove the intermediate claim Hb'subb: b' b.
An exact proof term for the current goal is (andER (x b') (b' b) Hb'prop).
We use b' to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb'B.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb'.
An exact proof term for the current goal is (Subq_tra b' b U Hb'subb HbsubU).
We will prove generated_topology X B generated_topology X B'.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U HU)).
We prove the intermediate claim HUprop: ∀xU, ∃b'B', x b' b' U.
An exact proof term for the current goal is (HRefProp U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃b0B', x0 b0 b0 U0) U (PowerI X U HUsubX) HUprop).
Theorem. (basis_finer_equiv_condition)
∀X B B' : set, basis_on X Bbasis_on X B'((∀xX, ∀bB, x b∃b'B', x b' b' b) finer_than (generated_topology X B') (generated_topology X B))
Proof:
Let X, B and B' be given.
Assume HB HB'.
Apply iffI to the current goal.
Assume Hcond.
An exact proof term for the current goal is (finer_via_basis X B B' HB HB' Hcond).
Assume Hfiner.
Let x be given.
Assume HxX.
Let b be given.
Assume HbB Hxb.
We prove the intermediate claim HbGen: b generated_topology X B.
An exact proof term for the current goal is (generated_topology_contains_basis X B HB b HbB).
We prove the intermediate claim HbGen': b generated_topology X B'.
An exact proof term for the current goal is (Hfiner b HbGen).
We prove the intermediate claim Hbprop: ∀x0b, ∃b'B', x0 b' b' b.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃b0B', x0 b0 b0 U0) b HbGen').
An exact proof term for the current goal is (Hbprop x Hxb).
Theorem. (generated_topology_finer)
∀X B T : set, basis_on X Btopology_on X T(∀bB, b T)finer_than T (generated_topology X B)
Proof:
Let X, B and T be given.
Assume HBasis HT HBsub.
We prove the intermediate claim HUnionClosed: ∀Fam𝒫 T, Fam T.
An exact proof term for the current goal is (andER ((T 𝒫 X Empty T) X T) (∀Fam𝒫 T, Fam T) (andEL (((T 𝒫 X Empty T) X T) (∀Fam𝒫 T, Fam T)) (∀UT, ∀VT, U V T) HT)).
We will prove generated_topology X B T.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U HU)).
We prove the intermediate claim HUprop: ∀xU, ∃bB, x b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U HU).
Set Fam to be the term {bB|b U} of type set.
We prove the intermediate claim HFamPowB: Fam 𝒫 B.
Apply PowerI B Fam to the current goal.
Let b be given.
Assume HbFam.
An exact proof term for the current goal is (SepE1 B (λb0 : setb0 U) b HbFam).
We prove the intermediate claim HUnionEq: Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (SepE2 B (λb0 : setb0 U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We prove the intermediate claim HbT: b T.
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HbFam: b Fam.
An exact proof term for the current goal is (SepI B (λb0 : setb0 U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
We prove the intermediate claim HFamPowT: Fam 𝒫 T.
Apply PowerI T Fam to the current goal.
Let b be given.
Assume HbFam.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (SepE1 B (λb0 : setb0 U) b HbFam).
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HUnionT: Fam T.
An exact proof term for the current goal is (HUnionClosed Fam HFamPowT).
rewrite the current goal using HUnionEq (from right to left).
An exact proof term for the current goal is HUnionT.
Proof:
Let X, B and T be given.
Assume HBasis HT HBsub.
An exact proof term for the current goal is (generated_topology_finer X B T HBasis HT HBsub).
Proof:
Let X and B be given.
Assume HBasis.
Apply set_ext to the current goal.
Let U be given.
Assume HU.
We prove the intermediate claim HUopen: open_in X (generated_topology X B) U.
An exact proof term for the current goal is (andI (topology_on X (generated_topology X B)) (U generated_topology X B) (lemma_topology_from_basis X B HBasis) HU).
We prove the intermediate claim HexFam: ∃Fam𝒫 B, Fam = U.
An exact proof term for the current goal is (open_sets_as_unions_of_basis X B HBasis U HUopen).
Apply HexFam to the current goal.
Let Fam be given.
Assume HFampair.
We prove the intermediate claim HFamPow: Fam 𝒫 B.
An exact proof term for the current goal is (andEL (Fam 𝒫 B) ( Fam = U) HFampair).
We prove the intermediate claim HUnion: Fam = U.
An exact proof term for the current goal is (andER (Fam 𝒫 B) ( Fam = U) HFampair).
We prove the intermediate claim HUnionFam: Fam { Fam0|Fam0𝒫 B}.
An exact proof term for the current goal is (ReplI (𝒫 B) (λFam0 : set Fam0) Fam HFamPow).
rewrite the current goal using HUnion (from right to left).
An exact proof term for the current goal is HUnionFam.
Let U be given.
Assume HUUnion.
We prove the intermediate claim HexFamPowRaw: ∃Fam𝒫 B, U = Fam.
An exact proof term for the current goal is (ReplE (𝒫 B) (λFam0 : set Fam0) U HUUnion).
We prove the intermediate claim HexFamPow: ∃Fam𝒫 B, Fam = U.
Apply HexFamPowRaw to the current goal.
Let Fam be given.
Assume HFamPair.
We prove the intermediate claim HFamPow: Fam 𝒫 B.
An exact proof term for the current goal is (andEL (Fam 𝒫 B) (U = Fam) HFamPair).
We prove the intermediate claim HUnion: U = Fam.
An exact proof term for the current goal is (andER (Fam 𝒫 B) (U = Fam) HFamPair).
We use Fam to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HFamPow.
rewrite the current goal using HUnion (from right to left).
Use reflexivity.
We prove the intermediate claim HUopen: open_in X (generated_topology X B) U.
An exact proof term for the current goal is (basis_generates_open_sets X B HBasis U HexFamPow).
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U generated_topology X B) HUopen).
Definition. We define singleton_basis to be λX ⇒ {{x,x}|xX} of type setset.
Proof:
Let X be given.
We will prove ((singleton_basis X 𝒫 X (∀xX, ∃bsingleton_basis X, x b)) (∀b1singleton_basis X, ∀b2singleton_basis X, ∀x : set, x b1x b2∃b3singleton_basis X, x b3 b3 b1 b2)).
Apply andI to the current goal.
Apply andI to the current goal.
We will prove singleton_basis X 𝒫 X.
Let s be given.
Assume Hs.
Apply (ReplE_impred X (λx0 : set{x0,x0}) s Hs) to the current goal.
Let x be given.
Assume HxX Hseq.
rewrite the current goal using Hseq (from left to right).
Apply PowerI to the current goal.
Let y be given.
Assume Hy.
Apply (UPairE y x x Hy (y X)) to the current goal.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxX.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxX.
We will prove ∀xX, ∃bsingleton_basis X, x b.
Let x be given.
Assume HxX.
We use {x,x} to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is (ReplI X (λx0 : set{x0,x0}) x HxX).
Apply UPairI1 to the current goal.
We will prove ∀b1singleton_basis X, ∀b2singleton_basis X, ∀x : set, x b1x b2∃b3singleton_basis X, x b3 b3 b1 b2.
Let b1 be given.
Assume Hb1.
Let b2 be given.
Assume Hb2.
Let x be given.
Assume Hx1 Hx2.
Apply (ReplE_impred X (λx0 : set{x0,x0}) b1 Hb1) to the current goal.
Let x1 be given.
Assume Hx1X Hb1eq.
Apply (ReplE_impred X (λx0 : set{x0,x0}) b2 Hb2) to the current goal.
Let x2 be given.
Assume Hx2X Hb2eq.
We prove the intermediate claim Hx1in: x {x1,x1}.
rewrite the current goal using Hb1eq (from right to left).
An exact proof term for the current goal is Hx1.
We prove the intermediate claim Hx2in: x {x2,x2}.
rewrite the current goal using Hb2eq (from right to left).
An exact proof term for the current goal is Hx2.
We prove the intermediate claim Hx_eq_x1: x = x1.
Apply (UPairE x x1 x1 Hx1in (x = x1)) to the current goal.
Assume Hxx1.
An exact proof term for the current goal is Hxx1.
Assume Hxx1.
An exact proof term for the current goal is Hxx1.
We prove the intermediate claim Hx_eq_x2: x = x2.
Apply (UPairE x x2 x2 Hx2in (x = x2)) to the current goal.
Assume Hxx2.
An exact proof term for the current goal is Hxx2.
Assume Hxx2.
An exact proof term for the current goal is Hxx2.
We prove the intermediate claim HxX: x X.
rewrite the current goal using Hx_eq_x1 (from left to right).
An exact proof term for the current goal is Hx1X.
We use {x,x} to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is (ReplI X (λx0 : set{x0,x0}) x HxX).
Apply andI to the current goal.
Apply UPairI1 to the current goal.
We will prove {x,x} b1 b2.
Let y be given.
Assume Hy.
Apply (UPairE y x x Hy (y b1 b2)) to the current goal.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
Apply binintersectI to the current goal.
An exact proof term for the current goal is Hx1.
An exact proof term for the current goal is Hx2.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
Apply binintersectI to the current goal.
An exact proof term for the current goal is Hx1.
An exact proof term for the current goal is Hx2.
Proof:
Let X be given.
Apply set_ext to the current goal.
Let U be given.
Assume HUgen.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃bsingleton_basis X, x0 b b U0) U HUgen).
Let U be given.
Assume HUinPow: U 𝒫 X.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U HUinPow).
We prove the intermediate claim HUprop: ∀xU, ∃bsingleton_basis X, x b b U.
Let x be given.
Assume HxU.
We use {x,x} to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is (ReplI X (λx0 : set{x0,x0}) x (HUsubX x HxU)).
Apply andI to the current goal.
An exact proof term for the current goal is (UPairI1 x x).
Let y be given.
Assume Hy.
Apply (UPairE y x x Hy (y U)) to the current goal.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxU.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃bsingleton_basis X, x0 b b U0) U HUinPow HUprop).
Definition. We define OrderedPair to be λx y ⇒ UPair x (UPair x y) of type setsetset.
Definition. We define R to be real of type set.
Definition. We define Q to be {qR|∃p n : set, p ω n ω q R} of type set.
Definition. We define Rlt to be λa b ⇒ a R b R a < b of type setsetprop.
Definition. We define EuclidPlane to be OrderedPair R R of type set.
Definition. We define distance_R2 to be λp c ⇒ Eps_i (λr ⇒ r R) of type setsetset.
Definition. We define circular_regions to be {U𝒫 EuclidPlane|∃c : set, ∃r : set, c EuclidPlane r R ¬ (r = 0) U = {pEuclidPlane|Rlt (distance_R2 p c) r}} of type set.
Definition. We define rectangular_regions to be {U𝒫 EuclidPlane|∃a b c d : set, a R b R c R d R U = {pEuclidPlane|∃x y : set, p = OrderedPair x y Rlt a x Rlt x b Rlt c y Rlt y d}} of type set.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
Let X, B and B' be given.
Assume HBasis Href.
We prove the intermediate claim Hprop: ∀Ugenerated_topology X B, ∀xU, ∃b'B', x b' b' U.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (∀Ugenerated_topology X B, ∀xU, ∃b'B', x b' b' U) Href).
We will prove generated_topology X B generated_topology X B'.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U HU)).
We prove the intermediate claim HUprop: ∀xU, ∃b'B', x b' b' U.
An exact proof term for the current goal is (Hprop U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃b0B', x0 b0 b0 U0) U (PowerI X U HUsubX) HUprop).
Definition. We define subbasis_on to be λX S ⇒ S 𝒫 X of type setsetprop.
Definition. We define intersection_of_family to be λFam ⇒ {x Fam|∀U : set, U Famx U} of type setset.
Definition. We define finite_subcollections to be λS ⇒ {F𝒫 S|finite F} of type setset.
Definition. We define finite_intersections_of to be λS ⇒ {intersection_of_family F|Ffinite_subcollections S} of type setset.
Definition. We define basis_of_subbasis to be λ_ S ⇒ {bfinite_intersections_of S|b Empty} of type setsetset.
Axiom. (finite_intersection_in_basis) We take the following as an axiom:
Axiom. (subbasis_elem_in_basis) We take the following as an axiom:
∀X S s : set, s Ss Emptys basis_of_subbasis X S
Definition. We define generated_topology_from_subbasis to be λX S ⇒ generated_topology X (basis_of_subbasis X S) of type setsetset.
Proof:
Let X and S be given.
Assume HS.
We will prove basis_on X (basis_of_subbasis X S).
The rest of this subproof is missing.
Proof:
Let X and S be given.
Assume HS.
The rest of this subproof is missing.
Proof:
Let X, S and T be given.
Assume HS HT HST.
The rest of this subproof is missing.
Theorem. (ex13_1_local_open_subset)
∀X T A : set, topology_on X T(∀xA, ∃UT, x U U A)open_in X T A
Proof:
Let X, T and A be given.
Assume HT Hlocal.
We will prove open_in X T A.
The rest of this subproof is missing.
Definition. We define a_elt to be Empty of type set.
Definition. We define b_elt to be {Empty} of type set.
Definition. We define c_elt to be {{Empty}} of type set.
Definition. We define abc_set to be UPair a_elt (UPair b_elt c_elt) of type set.
Definition. We define top_abc_1 to be UPair Empty abc_set of type set.
Definition. We define top_abc_2 to be 𝒫 abc_set of type set.
Definition. We define top_abc_3 to be UPair Empty (UPair {a_elt} abc_set) of type set.
Definition. We define top_abc_4 to be UPair Empty (UPair {b_elt} abc_set) of type set.
Definition. We define top_abc_5 to be UPair Empty (UPair {c_elt} abc_set) of type set.
Definition. We define top_abc_6 to be UPair Empty (UPair (UPair a_elt b_elt) abc_set) of type set.
Definition. We define top_abc_7 to be UPair Empty (UPair (UPair a_elt c_elt) abc_set) of type set.
Definition. We define top_abc_8 to be UPair Empty (UPair (UPair b_elt c_elt) abc_set) of type set.
Definition. We define top_abc_9 to be UPair Empty (UPair {a_elt} (UPair (UPair a_elt b_elt) abc_set)) of type set.
Definition. We define Intersection_Fam to be λFam ⇒ {U𝒫 ( Fam)|∀T : set, T FamU T} of type setset.
Proof:
Let Fam be given.
Let x be given.
Assume Hx.
An exact proof term for the current goal is (SepE1 ( Fam) (λx0 ⇒ ∀U : set, U Famx0 U) x Hx).
Proof:
Let Fam be given.
We will prove Empty Intersection_Fam Fam.
The rest of this subproof is missing.
Definition. We define infinite_complement_family to be λX ⇒ {U𝒫 X|infinite (X U) U = Empty U = X} of type setset.
Proof:
Let X be given.
The rest of this subproof is missing.
Proof:
Let X and U be given.
Assume Hnemp: U Empty.
We will prove countable (X U).
We prove the intermediate claim Hprop: countable (X U) U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λV : setcountable (X V) V = Empty) U HU).
We prove the intermediate claim Hcount_branch: countable (X U)countable (X U).
Assume Hcount.
An exact proof term for the current goal is Hcount.
We prove the intermediate claim Hempty_branch: U = Emptycountable (X U).
Assume HUeq.
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hnemp HUeq).
An exact proof term for the current goal is (Hprop (countable (X U)) Hcount_branch Hempty_branch).
Proof:
Let X and UFam be given.
Assume HUFam: UFam 𝒫 (countable_complement_topology X).
We will prove UFam countable_complement_topology X.
The rest of this subproof is missing.
Proof:
Let X be given.
The rest of this subproof is missing.
Proof:
Let X be given.
We will prove False.
The rest of this subproof is missing.
Proof:
Let X be given.
The rest of this subproof is missing.
Proof:
Let X and Fam be given.
Assume HfamTop: ∀TFam, topology_on X T.
We will prove topology_on X (Intersection_Fam Fam).
The rest of this subproof is missing.
Theorem. (ex13_4b_smallest_largest)
∀X Fam : set, ∃Tmin, topology_on X Tmin (∀TFam, T Tmin) (∀T', topology_on X T' (∀TFam, T T')Tmin T') ∃Tmax, topology_on X Tmax (∀TFam, Tmax T) (∀T', topology_on X T' (∀TFam, T' T)T' Tmax)
Proof:
Let X and Fam be given.
We will prove ∃Tmin, topology_on X Tmin (∀TFam, T Tmin) (∀T', topology_on X T' (∀TFam, T T')Tmin T') ∃Tmax, topology_on X Tmax (∀TFam, Tmax T) (∀T', topology_on X T' (∀TFam, T' T)T' Tmax).
The rest of this subproof is missing.
Proof:
We will prove ∃Tsmall Tall : set, topology_on abc_set Tsmall topology_on abc_set Tall.
The rest of this subproof is missing.
Proof:
Let X and A be given.
Assume HA: basis_on X A.
The rest of this subproof is missing.
Definition. We define rational_numbers to be ω of type set.
Definition. We define open_interval to be λa b ⇒ {xR|Rlt a x Rlt x b} of type setsetset.
Definition. We define halfopen_interval_left to be λa b ⇒ {xR|Rlt a x ¬ (Rlt b x)} of type setsetset.
Definition. We define R_standard_basis to be aR{open_interval a b|bR} of type set.
Definition. We define R_standard_topology to be generated_topology R R_standard_basis of type set.
Definition. We define R_lower_limit_basis to be aR{halfopen_interval_left a b|bR} of type set.
Definition. We define R_lower_limit_topology to be generated_topology R R_lower_limit_basis of type set.
Definition. We define inv_nat to be λn ⇒ n of type setset.
Axiom. (inv_nat_real) We take the following as an axiom:
∀n : set, n ωinv_nat n R
Definition. We define K_set to be {inv_nat n|nω} of type set.
Definition. We define R_K_basis to be aR{open_interval a b K_set|bR} of type set.
Definition. We define R_K_topology to be generated_topology R (R_standard_basis R_K_basis) of type set.
Proof:
The rest of this subproof is missing.
Definition. We define R_finite_complement_topology to be countable_complement_topology R of type set.
Definition. We define R_upper_limit_topology to be R_lower_limit_topology of type set.
Definition. We define R_ray_topology to be {U𝒫 R|U = Empty U = R (∃aR, {xR|Rlt a x} U)} of type set.
Definition. We define rational_open_intervals_basis to be q1rational_numbers{open_interval q1 q2|q2rational_numbers} of type set.
Definition. We define rational_halfopen_intervals_basis to be q1rational_numbers{halfopen_interval_left q1 q2|q2rational_numbers} of type set.
Definition. We define order_rel to be λX a b ⇒ (X = R Rlt a b) (X = OrderedPair R R ∃a1 a2 b1 b2 : set, a = OrderedPair a1 a2 b = OrderedPair b1 b2 (Rlt a1 b1 (a1 = b1 Rlt a2 b2))) of type setsetsetprop.
Definition. We define order_topology_basis to be λX ⇒ ({I𝒫 X|∃aX, ∃bX, I = {xX|order_rel X a x order_rel X x b}} {I𝒫 X|∃bX, I = {xX|order_rel X x b}} {I𝒫 X|∃aX, I = {xX|order_rel X a x}}) of type setset.
Definition. We define order_topology to be λX ⇒ generated_topology X (order_topology_basis X) of type setset.
Proof:
Let X be given.
We will prove topology_on X (order_topology X).
The rest of this subproof is missing.
Definition. We define open_ray_upper to be λX a ⇒ {xX|order_rel X a x} of type setsetset.
Definition. We define open_ray_lower to be λX a ⇒ {xX|order_rel X x a} of type setsetset.
Proof:
Let X be given.
We will prove ∃S : set, generated_topology X S = order_topology X.
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define R2_dictionary_order_topology to be order_topology (OrderedPair R R) of type set.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define Zplus to be ω of type set.
Proof:
The rest of this subproof is missing.
Definition. We define two_by_nat to be OrderedPair 2 ω of type set.
Definition. We define two_by_nat_order_topology to be order_topology two_by_nat of type set.
Proof:
The rest of this subproof is missing.
Definition. We define rectangle_set to be λU V ⇒ OrderedPair U V of type setsetset.
Definition. We define product_subbasis to be λX Tx Y Ty ⇒ UTx{rectangle_set U V|VTy} of type setsetsetsetset.
Definition. We define product_topology to be λX Tx Y Ty ⇒ generated_topology (OrderedPair X Y) (product_subbasis X Tx Y Ty) of type setsetsetsetset.
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove topology_on (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
Theorem. (product_basis_generates)
∀X Tx Y Ty Bx By : set, basis_on X Bx generated_topology X Bx = Txbasis_on Y By generated_topology Y By = Ty∃B : set, basis_on (OrderedPair X Y) B (∀UBx, ∀VBy, OrderedPair U V B) generated_topology (OrderedPair X Y) B = product_topology X Tx Y Ty
Proof:
Let X, Tx, Y, Ty, Bx and By be given.
Assume HBx: basis_on X Bx generated_topology X Bx = Tx.
Assume HBy: basis_on Y By generated_topology Y By = Ty.
We will prove ∃B : set, basis_on (OrderedPair X Y) B (∀UBx, ∀VBy, OrderedPair U V B) generated_topology (OrderedPair X Y) B = product_topology X Tx Y Ty.
The rest of this subproof is missing.
Definition. We define projection1 to be λX Y ⇒ {p𝒫 (OrderedPair (OrderedPair X Y) X)|∃x : set, ∃y : set, x X y Y p = UPair (OrderedPair x y) x} of type setsetset.
Definition. We define projection2 to be λX Y ⇒ {p𝒫 (OrderedPair (OrderedPair X Y) Y)|∃x : set, ∃y : set, x X y Y p = UPair (OrderedPair x y) y} of type setsetset.
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove ∃S : set, S = product_subbasis X Tx Y Ty generated_topology (OrderedPair X Y) S = product_topology X Tx Y Ty.
The rest of this subproof is missing.
Definition. We define apply_fun to be λf x ⇒ Eps_i (λy ⇒ UPair x y f) of type setsetset.
Definition. We define function_on to be λf X Y ⇒ ∀x : set, x Xapply_fun f x Y of type setsetsetprop.
Definition. We define function_space to be λX Y ⇒ {f𝒫 (OrderedPair X Y)|function_on f X Y} of type setsetset.
Definition. We define const_family to be λI X ⇒ {UPair i X|iI} of type setsetset.
Definition. We define product_component to be λXi i ⇒ apply_fun Xi i of type setsetset.
Definition. We define product_component_topology to be λXi i ⇒ apply_fun Xi i of type setsetset.
Definition. We define product_space to be λI Xi ⇒ {f𝒫 ( Xi)|function_on f I ( Xi) ∀i : set, i Iapply_fun f i apply_fun Xi i} of type setsetset.
Definition. We define product_cylinder to be λI Xi i U ⇒ {fproduct_space I Xi|i I U apply_fun Xi i apply_fun f i U} of type setsetsetsetset.
Definition. We define product_subbasis_full to be λI Xi ⇒ iI{product_cylinder I Xi i U|Uapply_fun Xi i} of type setsetset.
Definition. We define product_topology_full to be λI Xi ⇒ generated_topology (product_space I Xi) (product_subbasis_full I Xi) of type setsetset.
Definition. We define box_topology to be λI Xi ⇒ generated_topology (product_space I Xi) (𝒫 (product_space I Xi)) of type setsetset.
Definition. We define countable_product_space to be λI Xi ⇒ product_space I Xi of type setsetset.
Definition. We define countable_product_topology to be λI Xi ⇒ product_topology_full I Xi of type setsetset.
Definition. We define euclidean_space to be λn ⇒ product_space n (const_family n R) of type setset.
Definition. We define euclidean_topology to be λn ⇒ product_topology_full n (const_family n R) of type setset.
Definition. We define R2_standard_topology to be product_topology R R_standard_topology R R_standard_topology of type set.
Proof:
The rest of this subproof is missing.
Definition. We define subspace_topology to be λX Tx Y ⇒ {U𝒫 Y|∃VTx, U = V Y} of type setsetsetset.
Proof:
Let X, Tx and Y be given.
Assume HTx: topology_on X Tx.
Assume HY: Y X.
We will prove topology_on Y (subspace_topology X Tx Y).
Apply andI to the current goal.
We will prove (subspace_topology X Tx Y 𝒫 Y Empty subspace_topology X Tx Y) Y subspace_topology X Tx Y (∀UFam𝒫 (subspace_topology X Tx Y), UFam subspace_topology X Tx Y).
Apply andI to the current goal.
Apply andI to the current goal.
We will prove subspace_topology X Tx Y 𝒫 Y Empty subspace_topology X Tx Y.
Apply andI to the current goal.
We will prove subspace_topology X Tx Y 𝒫 Y.
Let U be given.
Assume HU: U subspace_topology X Tx Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HU).
We will prove Empty subspace_topology X Tx Y.
We prove the intermediate claim HEmptyTx: Empty Tx.
An exact proof term for the current goal is (topology_has_empty X Tx HTx).
We prove the intermediate claim HPred: ∃VTx, Empty = V Y.
We use Empty to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HEmptyTx.
We will prove Empty = Empty Y.
We prove the intermediate claim H1: Empty Y = Empty.
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_1 Empty Y).
rewrite the current goal using H1 (from left to right).
Use reflexivity.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) Empty (Empty_In_Power Y) HPred).
We will prove Y subspace_topology X Tx Y.
We prove the intermediate claim HXTx: X Tx.
An exact proof term for the current goal is (topology_has_X X Tx HTx).
We prove the intermediate claim HPredY: ∃VTx, Y = V Y.
We use X to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HXTx.
We will prove Y = X Y.
Apply set_ext to the current goal.
Let y be given.
Assume Hy: y Y.
Apply binintersectI to the current goal.
An exact proof term for the current goal is (HY y Hy).
An exact proof term for the current goal is Hy.
An exact proof term for the current goal is (binintersect_Subq_2 X Y).
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) Y (Self_In_Power Y) HPredY).
We will prove ∀UFam𝒫 (subspace_topology X Tx Y), UFam subspace_topology X Tx Y.
Let UFam be given.
Assume HUFam: UFam 𝒫 (subspace_topology X Tx Y).
We will prove UFam subspace_topology X Tx Y.
We prove the intermediate claim HUFamsub: UFam subspace_topology X Tx Y.
An exact proof term for the current goal is (PowerE (subspace_topology X Tx Y) UFam HUFam).
Set VFam to be the term {VTx|∃UUFam, U = V Y}.
We prove the intermediate claim HVFamTx: VFam Tx.
Let V be given.
Assume HV: V VFam.
An exact proof term for the current goal is (SepE1 Tx (λV0 ⇒ ∃UUFam, U = V0 Y) V HV).
We prove the intermediate claim HVFamPower: VFam 𝒫 Tx.
Apply PowerI to the current goal.
An exact proof term for the current goal is HVFamTx.
We prove the intermediate claim HUnionVFam: VFam Tx.
An exact proof term for the current goal is (topology_union_closed X Tx VFam HTx HVFamTx).
We prove the intermediate claim HUnionEq: UFam = ( VFam) Y.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x UFam.
Apply UnionE_impred UFam x Hx to the current goal.
Let U be given.
Assume HxU: x U.
Assume HUinFam: U UFam.
We prove the intermediate claim HUinSubspace: U subspace_topology X Tx Y.
An exact proof term for the current goal is (HUFamsub U HUinFam).
We prove the intermediate claim HUexists: ∃VTx, U = V Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HUinSubspace).
Apply HUexists to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HVTx: V Tx.
Assume HUeq: U = V Y.
Apply binintersectI to the current goal.
We will prove x VFam.
We prove the intermediate claim HxV: x V.
We prove the intermediate claim HxVY: x V Y.
rewrite the current goal using HUeq (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (binintersectE1 V Y x HxVY).
We prove the intermediate claim HVinVFam: V VFam.
Apply (SepI Tx (λV0 ⇒ ∃UUFam, U = V0 Y) V HVTx) to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUinFam.
An exact proof term for the current goal is HUeq.
An exact proof term for the current goal is (UnionI VFam x V HxV HVinVFam).
We will prove x Y.
We prove the intermediate claim HxVY: x V Y.
rewrite the current goal using HUeq (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (binintersectE2 V Y x HxVY).
Let x be given.
Assume Hx: x ( VFam) Y.
We prove the intermediate claim HxUnionV: x VFam.
An exact proof term for the current goal is (binintersectE1 ( VFam) Y x Hx).
We prove the intermediate claim HxY: x Y.
An exact proof term for the current goal is (binintersectE2 ( VFam) Y x Hx).
Apply UnionE_impred VFam x HxUnionV to the current goal.
Let V be given.
Assume HxV: x V.
Assume HVinVFam: V VFam.
We prove the intermediate claim HVexists: ∃UUFam, U = V Y.
An exact proof term for the current goal is (SepE2 Tx (λV0 ⇒ ∃UUFam, U = V0 Y) V HVinVFam).
Apply HVexists to the current goal.
Let U be given.
Assume HUandEq.
Apply HUandEq to the current goal.
Assume HUinFam: U UFam.
Assume HUeq: U = V Y.
We prove the intermediate claim HxU: x U.
rewrite the current goal using HUeq (from left to right).
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV.
An exact proof term for the current goal is HxY.
An exact proof term for the current goal is (UnionI UFam x U HxU HUinFam).
We prove the intermediate claim HPredUnion: ∃VTx, UFam = V Y.
We use ( VFam) to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUnionVFam.
An exact proof term for the current goal is HUnionEq.
We prove the intermediate claim HUnionInPowerY: UFam 𝒫 Y.
Apply PowerI to the current goal.
Let x be given.
Assume Hx: x UFam.
We prove the intermediate claim HxVY: x ( VFam) Y.
rewrite the current goal using HUnionEq (from right to left).
An exact proof term for the current goal is Hx.
An exact proof term for the current goal is (binintersectE2 ( VFam) Y x HxVY).
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) ( UFam) HUnionInPowerY HPredUnion).
We will prove ∀Usubspace_topology X Tx Y, ∀Vsubspace_topology X Tx Y, U V subspace_topology X Tx Y.
Let U be given.
Assume HU: U subspace_topology X Tx Y.
Let V be given.
Assume HV: V subspace_topology X Tx Y.
We will prove U V subspace_topology X Tx Y.
We prove the intermediate claim HUexists: ∃V1Tx, U = V1 Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HU).
We prove the intermediate claim HVexists: ∃V2Tx, V = V2 Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λV0 : set∃VTx, V0 = V Y) V HV).
Apply HUexists to the current goal.
Let V1 be given.
Assume HV1andEq.
Apply HV1andEq to the current goal.
Assume HV1Tx: V1 Tx.
Assume HUeq: U = V1 Y.
Apply HVexists to the current goal.
Let V2 be given.
Assume HV2andEq.
Apply HV2andEq to the current goal.
Assume HV2Tx: V2 Tx.
Assume HVeq: V = V2 Y.
We prove the intermediate claim HV1V2: V1 V2 Tx.
An exact proof term for the current goal is (topology_binintersect_closed X Tx V1 V2 HTx HV1Tx HV2Tx).
We prove the intermediate claim HIntEq: U V = (V1 V2) Y.
rewrite the current goal using HUeq (from left to right).
rewrite the current goal using HVeq (from left to right).
We will prove (V1 Y) (V2 Y) = (V1 V2) Y.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x (V1 Y) (V2 Y).
We prove the intermediate claim HxV1Y: x V1 Y.
An exact proof term for the current goal is (binintersectE1 (V1 Y) (V2 Y) x Hx).
We prove the intermediate claim HxV2Y: x V2 Y.
An exact proof term for the current goal is (binintersectE2 (V1 Y) (V2 Y) x Hx).
We prove the intermediate claim HxV1: x V1.
An exact proof term for the current goal is (binintersectE1 V1 Y x HxV1Y).
We prove the intermediate claim HxV2: x V2.
An exact proof term for the current goal is (binintersectE1 V2 Y x HxV2Y).
We prove the intermediate claim HxY: x Y.
An exact proof term for the current goal is (binintersectE2 V1 Y x HxV1Y).
Apply binintersectI to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV1.
An exact proof term for the current goal is HxV2.
An exact proof term for the current goal is HxY.
Let x be given.
Assume Hx: x (V1 V2) Y.
We prove the intermediate claim HxV1V2: x V1 V2.
An exact proof term for the current goal is (binintersectE1 (V1 V2) Y x Hx).
We prove the intermediate claim HxY: x Y.
An exact proof term for the current goal is (binintersectE2 (V1 V2) Y x Hx).
We prove the intermediate claim HxV1: x V1.
An exact proof term for the current goal is (binintersectE1 V1 V2 x HxV1V2).
We prove the intermediate claim HxV2: x V2.
An exact proof term for the current goal is (binintersectE2 V1 V2 x HxV1V2).
Apply binintersectI to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV1.
An exact proof term for the current goal is HxY.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV2.
An exact proof term for the current goal is HxY.
We prove the intermediate claim HPredInt: ∃WTx, U V = W Y.
We use (V1 V2) to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV1V2.
An exact proof term for the current goal is HIntEq.
We prove the intermediate claim HIntInPowerY: U V 𝒫 Y.
Apply PowerI to the current goal.
Let x be given.
Assume Hx: x U V.
We prove the intermediate claim HxU: x U.
An exact proof term for the current goal is (binintersectE1 U V x Hx).
We prove the intermediate claim HUinPowerY: U 𝒫 Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HU).
We prove the intermediate claim HUsub: U Y.
An exact proof term for the current goal is (PowerE Y U HUinPowerY).
An exact proof term for the current goal is (HUsub x HxU).
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) (U V) HIntInPowerY HPredInt).
Theorem. (open_in_subspace_iff)
∀X Tx Y U : set, topology_on X TxY XU Y(open_in Y (subspace_topology X Tx Y) U ∃VTx, U = V Y)
Proof:
Let X, Tx, Y and U be given.
Assume HTx: topology_on X Tx.
Assume HY: Y X.
Assume HU: U Y.
We will prove open_in Y (subspace_topology X Tx Y) U ∃VTx, U = V Y.
Apply iffI to the current goal.
Assume HopenU: open_in Y (subspace_topology X Tx Y) U.
We will prove ∃VTx, U = V Y.
We prove the intermediate claim HUinSubspace: U subspace_topology X Tx Y.
An exact proof term for the current goal is (andER (topology_on Y (subspace_topology X Tx Y)) (U subspace_topology X Tx Y) HopenU).
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HUinSubspace).
Assume Hexists: ∃VTx, U = V Y.
We will prove open_in Y (subspace_topology X Tx Y) U.
We will prove topology_on Y (subspace_topology X Tx Y) U subspace_topology X Tx Y.
Apply andI to the current goal.
We will prove topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HY).
We will prove U subspace_topology X Tx Y.
We prove the intermediate claim HUinPowerY: U 𝒫 Y.
Apply PowerI to the current goal.
An exact proof term for the current goal is HU.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HUinPowerY Hexists).
Proof:
Let X, Tx, Y and B be given.
Assume HTx: topology_on X Tx.
Assume HB: basis_on X B generated_topology X B = Tx.
The rest of this subproof is missing.
Theorem. (open_in_subspace_if_ambient_open)
∀X Tx Y U : set, topology_on X TxY TxU Yopen_in Y (subspace_topology X Tx Y) UU Tx
Proof:
Let X, Tx, Y and U be given.
Assume HTx: topology_on X Tx.
Assume HY: Y Tx.
Assume HU: U Y.
Assume HUopen: open_in Y (subspace_topology X Tx Y) U.
We will prove U Tx.
We prove the intermediate claim HYsub: Y X.
An exact proof term for the current goal is (topology_elem_subset X Tx Y HTx HY).
We prove the intermediate claim HUiffExists: open_in Y (subspace_topology X Tx Y) U ∃VTx, U = V Y.
An exact proof term for the current goal is (open_in_subspace_iff X Tx Y U HTx HYsub HU).
We prove the intermediate claim Hexists: ∃VTx, U = V Y.
An exact proof term for the current goal is (iffEL (open_in Y (subspace_topology X Tx Y) U) (∃VTx, U = V Y) HUiffExists HUopen).
Apply Hexists to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V Tx.
Assume HUeq: U = V Y.
We prove the intermediate claim HVY: V Y Tx.
An exact proof term for the current goal is (topology_binintersect_closed X Tx V Y HTx HV HY).
We prove the intermediate claim HUinTx: U Tx.
rewrite the current goal using HUeq (from left to right).
An exact proof term for the current goal is HVY.
An exact proof term for the current goal is HUinTx.
Theorem. (product_subspace_topology)
∀X Tx Y Ty A B : set, topology_on X Txtopology_on Y TyA XB Yproduct_topology A (subspace_topology X Tx A) B (subspace_topology Y Ty B) = subspace_topology (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B)
Proof:
Let X, Tx, Y, Ty, A and B be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
Assume HA: A X.
Assume HB: B Y.
The rest of this subproof is missing.
Definition. We define unit_interval to be R of type set.
Definition. We define ordered_square to be OrderedPair unit_interval unit_interval of type set.
Definition. We define ordered_square_topology to be order_topology ordered_square of type set.
Definition. We define ordered_square_open_strip to be ordered_square of type set.
Definition. We define ordered_square_subspace_topology to be subspace_topology (OrderedPair R R) R2_dictionary_order_topology ordered_square of type set.
Proof:
The rest of this subproof is missing.
Proof:
Let X and Y be given.
The rest of this subproof is missing.
Theorem. (binintersect_right_absorb_subset)
∀W Y A : set, A Y(W Y) A = W A
Proof:
Let W, Y and A be given.
Assume Hsub: A Y.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x (W Y) A.
We prove the intermediate claim Hpair: x W Y x A.
An exact proof term for the current goal is (binintersectE (W Y) A x Hx).
We prove the intermediate claim HWY: x W Y.
An exact proof term for the current goal is (andEL (x W Y) (x A) Hpair).
We prove the intermediate claim HA: x A.
An exact proof term for the current goal is (andER (x W Y) (x A) Hpair).
We prove the intermediate claim HWYpair: x W x Y.
An exact proof term for the current goal is (binintersectE W Y x HWY).
We prove the intermediate claim HW: x W.
An exact proof term for the current goal is (andEL (x W) (x Y) HWYpair).
Apply binintersectI to the current goal.
An exact proof term for the current goal is HW.
An exact proof term for the current goal is HA.
Let x be given.
Assume Hx: x W A.
We prove the intermediate claim Hpair: x W x A.
An exact proof term for the current goal is (binintersectE W A x Hx).
We prove the intermediate claim HW: x W.
An exact proof term for the current goal is (andEL (x W) (x A) Hpair).
We prove the intermediate claim HA: x A.
An exact proof term for the current goal is (andER (x W) (x A) Hpair).
We prove the intermediate claim HY: x Y.
An exact proof term for the current goal is (Hsub x HA).
We prove the intermediate claim HWY: x W Y.
An exact proof term for the current goal is (binintersectI W Y x HW HY).
Apply binintersectI to the current goal.
An exact proof term for the current goal is HWY.
An exact proof term for the current goal is HA.
Theorem. (ex16_1_subspace_transitive)
∀X Tx Y A : set, topology_on X TxY XA Ysubspace_topology Y (subspace_topology X Tx Y) A = subspace_topology X Tx A
Proof:
Let X, Tx, Y and A be given.
Assume Htop: topology_on X Tx.
Assume HY: Y X.
Assume HA: A Y.
We will prove subspace_topology Y (subspace_topology X Tx Y) A = subspace_topology X Tx A.
Apply set_ext to the current goal.
Let W be given.
Assume HW: W subspace_topology Y (subspace_topology X Tx Y) A.
We will prove W subspace_topology X Tx A.
We prove the intermediate claim HWPowerA: W 𝒫 A.
An exact proof term for the current goal is (SepE1 (𝒫 A) (λU0 : set∃Usubspace_topology X Tx Y, U0 = U A) W HW).
We prove the intermediate claim HWexists: ∃Usubspace_topology X Tx Y, W = U A.
An exact proof term for the current goal is (SepE2 (𝒫 A) (λU0 : set∃Usubspace_topology X Tx Y, U0 = U A) W HW).
Apply HWexists to the current goal.
Let U be given.
Assume HU: U subspace_topology X Tx Y W = U A.
We prove the intermediate claim HUinSubY: U subspace_topology X Tx Y.
An exact proof term for the current goal is (andEL (U subspace_topology X Tx Y) (W = U A) HU).
We prove the intermediate claim HWeqUA: W = U A.
An exact proof term for the current goal is (andER (U subspace_topology X Tx Y) (W = U A) HU).
We prove the intermediate claim HUPowerY: U 𝒫 Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HUinSubY).
We prove the intermediate claim HUexists: ∃VTx, U = V Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HUinSubY).
Apply HUexists to the current goal.
Let V be given.
Assume HV: V Tx U = V Y.
We prove the intermediate claim HVinTx: V Tx.
An exact proof term for the current goal is (andEL (V Tx) (U = V Y) HV).
We prove the intermediate claim HUeqVY: U = V Y.
An exact proof term for the current goal is (andER (V Tx) (U = V Y) HV).
We prove the intermediate claim HWeqVA: W = V A.
rewrite the current goal using HWeqUA (from left to right).
rewrite the current goal using HUeqVY (from left to right).
An exact proof term for the current goal is (binintersect_right_absorb_subset V Y A HA).
We prove the intermediate claim HWPred: ∃V0Tx, W = V0 A.
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HVinTx.
An exact proof term for the current goal is HWeqVA.
An exact proof term for the current goal is (SepI (𝒫 A) (λW0 : set∃V0Tx, W0 = V0 A) W HWPowerA HWPred).
Let W be given.
Assume HW: W subspace_topology X Tx A.
We will prove W subspace_topology Y (subspace_topology X Tx Y) A.
We prove the intermediate claim HWPowerA: W 𝒫 A.
An exact proof term for the current goal is (SepE1 (𝒫 A) (λW0 : set∃VTx, W0 = V A) W HW).
We prove the intermediate claim HWexists: ∃VTx, W = V A.
An exact proof term for the current goal is (SepE2 (𝒫 A) (λW0 : set∃VTx, W0 = V A) W HW).
Apply HWexists to the current goal.
Let V be given.
Assume HV: V Tx W = V A.
We prove the intermediate claim HVinTx: V Tx.
An exact proof term for the current goal is (andEL (V Tx) (W = V A) HV).
We prove the intermediate claim HWeqVA: W = V A.
An exact proof term for the current goal is (andER (V Tx) (W = V A) HV).
Set U to be the term V Y.
We prove the intermediate claim HUinSubY: U subspace_topology X Tx Y.
We prove the intermediate claim HUPowerY: U 𝒫 Y.
An exact proof term for the current goal is (PowerI Y U (binintersect_Subq_2 V Y)).
We prove the intermediate claim HUPred: ∃V0Tx, U = V0 Y.
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HVinTx.
Use reflexivity.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set∃V0Tx, U0 = V0 Y) U HUPowerY HUPred).
We prove the intermediate claim HWeqUA: W = U A.
rewrite the current goal using HWeqVA (from left to right).
Use symmetry.
An exact proof term for the current goal is (binintersect_right_absorb_subset V Y A HA).
We prove the intermediate claim HWPred: ∃U0subspace_topology X Tx Y, W = U0 A.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUinSubY.
An exact proof term for the current goal is HWeqUA.
An exact proof term for the current goal is (SepI (𝒫 A) (λW0 : set∃U0subspace_topology X Tx Y, W0 = U0 A) W HWPowerA HWPred).
Theorem. (ex16_2_finer_subspaces)
∀X T T' Y : set, topology_on X Ttopology_on X T'T' TY Xsubspace_topology X T' Y subspace_topology X T Y
Proof:
Let X, T, T' and Y be given.
Assume Htop: topology_on X T.
Assume Htop': topology_on X T'.
Assume Hfiner: T' T.
Assume HY: Y X.
We will prove subspace_topology X T' Y subspace_topology X T Y.
Let W be given.
Assume HW: W subspace_topology X T' Y.
We will prove W subspace_topology X T Y.
We prove the intermediate claim HWPowerY: W 𝒫 Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λW0 : set∃VT', W0 = V Y) W HW).
We prove the intermediate claim HWexists: ∃VT', W = V Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λW0 : set∃VT', W0 = V Y) W HW).
Apply HWexists to the current goal.
Let V' be given.
Assume HV': V' T' W = V' Y.
We prove the intermediate claim HV'inT': V' T'.
An exact proof term for the current goal is (andEL (V' T') (W = V' Y) HV').
We prove the intermediate claim HWeqV'Y: W = V' Y.
An exact proof term for the current goal is (andER (V' T') (W = V' Y) HV').
We prove the intermediate claim HV'inT: V' T.
An exact proof term for the current goal is (Hfiner V' HV'inT').
We prove the intermediate claim HWPred: ∃VT, W = V Y.
We use V' to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV'inT.
An exact proof term for the current goal is HWeqV'Y.
An exact proof term for the current goal is (SepI (𝒫 Y) (λW0 : set∃VT, W0 = V Y) W HWPowerY HWPred).
Definition. We define interval_A to be open_interval Empty (𝒫 Empty) of type set.
Definition. We define interval_B to be open_interval (𝒫 Empty) (𝒫 (𝒫 Empty)) of type set.
Definition. We define interval_C to be open_interval Empty Empty of type set.
Definition. We define interval_D to be open_interval (𝒫 Empty) (𝒫 Empty) of type set.
Definition. We define interval_E to be open_interval (𝒫 (𝒫 Empty)) (𝒫 (𝒫 Empty)) of type set.
Theorem. (ex16_3_open_sets_subspace)
∀X Tx Y : set, topology_on X TxY X∀U : set, open_in Y (subspace_topology X Tx Y) U∃V : set, open_in X Tx V U = V Y
Proof:
Let X, Tx and Y be given.
Assume Htop: topology_on X Tx.
Assume HY: Y X.
Let U be given.
Assume HU: open_in Y (subspace_topology X Tx Y) U.
We will prove ∃V : set, open_in X Tx V U = V Y.
We prove the intermediate claim HtopY: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (andEL (topology_on Y (subspace_topology X Tx Y)) (U subspace_topology X Tx Y) HU).
We prove the intermediate claim HUinSub: U subspace_topology X Tx Y.
An exact proof term for the current goal is (andER (topology_on Y (subspace_topology X Tx Y)) (U subspace_topology X Tx Y) HU).
We prove the intermediate claim HUPowerY: U 𝒫 Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HUinSub).
We prove the intermediate claim HUexists: ∃VTx, U = V Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HUinSub).
Apply HUexists to the current goal.
Let V be given.
Assume HV: V Tx U = V Y.
We prove the intermediate claim HVinTx: V Tx.
An exact proof term for the current goal is (andEL (V Tx) (U = V Y) HV).
We prove the intermediate claim HUeqVY: U = V Y.
An exact proof term for the current goal is (andER (V Tx) (U = V Y) HV).
We prove the intermediate claim HVopen: open_in X Tx V.
An exact proof term for the current goal is (andI (topology_on X Tx) (V Tx) Htop HVinTx).
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HVopen.
An exact proof term for the current goal is HUeqVY.
Definition. We define projection_image1 to be λX Y U ⇒ {xX|∃y : set, OrderedPair x y U} of type setsetsetset.
Definition. We define projection_image2 to be λX Y U ⇒ {yY|∃x : set, OrderedPair x y U} of type setsetsetset.
Theorem. (ex16_4_projections_open)
∀X Tx Y Ty : set, topology_on X Txtopology_on Y Ty∀U : set, U product_topology X Tx Y Tyopen_in X Tx (projection_image1 X Y U) open_in Y Ty (projection_image2 X Y U)
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
Let U be given.
Assume HU: U product_topology X Tx Y Ty.
We will prove open_in X Tx (projection_image1 X Y U) open_in Y Ty (projection_image2 X Y U).
The rest of this subproof is missing.
Theorem. (ex16_5a_product_monotone)
∀X T T' Y U U' : set, topology_on X Ttopology_on X T'topology_on Y Utopology_on Y U'T T' U U'product_topology X T Y U product_topology X T' Y U'
Proof:
Let X, T, T', Y, U and U' be given.
Assume HTx: topology_on X T.
Assume HTx': topology_on X T'.
Assume HTy: topology_on Y U.
Assume HTy': topology_on Y U'.
Assume Hfiner: T T' U U'.
We will prove product_topology X T Y U product_topology X T' Y U'.
The rest of this subproof is missing.
Theorem. (ex16_5b_product_converse)
∀X T T' Y U U' : set, topology_on X Ttopology_on X T'topology_on Y Utopology_on Y U'product_topology X T Y U product_topology X T' Y U'T T' U U'
Proof:
Let X, T, T', Y, U and U' be given.
Assume HTx: topology_on X T.
Assume HTx': topology_on X T'.
Assume HTy: topology_on Y U.
Assume HTy': topology_on Y U'.
Assume Hprod: product_topology X T Y U product_topology X T' Y U'.
We will prove T T' U U'.
The rest of this subproof is missing.
Definition. We define rational_rectangle_basis to be {r𝒫 (OrderedPair R R)|∃a b c d : set, a rational_numbers b rational_numbers c rational_numbers d rational_numbers r = OrderedPair (open_interval a b) (open_interval c d)} of type set.
Definition. We define convex_subset to be λA ⇒ A R ∀x y : set, x Ay Aopen_interval x y A of type setprop.
Proof:
Let A be given.
Assume HA: convex_subset A.
We will prove A = Empty A = R ∃a b : set, A = open_interval a b A = {xR|Rlt a x} A = {xR|Rlt x b}.
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define interior_of to be λX T A ⇒ {xX|∃U : set, U T x U U A} of type setsetsetset.
Definition. We define closure_of to be λX T A ⇒ {xX|∀U : set, U Tx UU A Empty} of type setsetsetset.
Theorem. (subset_of_closure)
∀X Tx A : set, topology_on X TxA XA closure_of X Tx A
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
Assume HA: A X.
We will prove A closure_of X Tx A.
Let x be given.
Assume Hx: x A.
We will prove x closure_of X Tx A.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (HA x Hx).
We prove the intermediate claim Hcond: ∀U : set, U Txx UU A Empty.
Let U be given.
Assume HU: U Tx.
Assume HxU: x U.
We will prove U A Empty.
Assume Hempty: U A = Empty.
We prove the intermediate claim HxUA: x U A.
An exact proof term for the current goal is (binintersectI U A x HxU Hx).
We prove the intermediate claim HxEmpty: x Empty.
rewrite the current goal using Hempty (from right to left).
An exact proof term for the current goal is HxUA.
An exact proof term for the current goal is (EmptyE x HxEmpty).
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U Txx0 UU A Empty) x HxX Hcond).
Theorem. (closure_monotone)
∀X Tx A B : set, topology_on X TxA BB Xclosure_of X Tx A closure_of X Tx B
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HAB: A B.
Assume HB: B X.
We will prove closure_of X Tx A closure_of X Tx B.
Let x be given.
Assume Hx: x closure_of X Tx A.
We will prove x closure_of X Tx B.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (SepE1 X (λx0 ⇒ ∀U : set, U Txx0 UU A Empty) x Hx).
We prove the intermediate claim HcondA: ∀U : set, U Txx UU A Empty.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∀U : set, U Txx0 UU A Empty) x Hx).
We prove the intermediate claim HcondB: ∀U : set, U Txx UU B Empty.
Let U be given.
Assume HU: U Tx.
Assume HxU: x U.
We will prove U B Empty.
We prove the intermediate claim HUA_ne: U A Empty.
An exact proof term for the current goal is (HcondA U HU HxU).
Assume Hempty: U B = Empty.
We prove the intermediate claim HUA_sub_UB: U A U B.
Let y be given.
Assume Hy: y U A.
We prove the intermediate claim HyU: y U.
An exact proof term for the current goal is (binintersectE1 U A y Hy).
We prove the intermediate claim HyA: y A.
An exact proof term for the current goal is (binintersectE2 U A y Hy).
We prove the intermediate claim HyB: y B.
An exact proof term for the current goal is (HAB y HyA).
An exact proof term for the current goal is (binintersectI U B y HyU HyB).
We prove the intermediate claim HUA_empty: U A = Empty.
Apply Empty_Subq_eq to the current goal.
We prove the intermediate claim HUB_sub_Empty: U B Empty.
rewrite the current goal using Hempty (from left to right).
An exact proof term for the current goal is (Subq_ref Empty).
An exact proof term for the current goal is (Subq_tra (U A) (U B) Empty HUA_sub_UB HUB_sub_Empty).
An exact proof term for the current goal is (HUA_ne HUA_empty).
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U Txx0 UU B Empty) x HxX HcondB).
Theorem. (closed_sets_axioms)
∀X T : set, topology_on X Tlet C ≔ {X U|UT} in X C Empty C (∀F : set, F 𝒫 Cintersection_of_family F C) (∀A B : set, A CB CA B C)
Proof:
Let X and T be given.
Assume HT: topology_on X T.
We will prove let C ≔ {X U|UT} in X C Empty C (∀F : set, F 𝒫 Cintersection_of_family F C) (∀A B : set, A CB CA B C).
The rest of this subproof is missing.
Theorem. (closed_in_subspace_iff_intersection)
∀X Tx Y A : set, topology_on X TxY X(closed_in Y (subspace_topology X Tx Y) A ∃C : set, closed_in X Tx C A = C Y)
Proof:
Let X, Tx, Y and A be given.
Assume HTx: topology_on X Tx.
Assume HY: Y X.
We will prove closed_in Y (subspace_topology X Tx Y) A ∃C : set, closed_in X Tx C A = C Y.
Apply iffI to the current goal.
Assume HAclosed: closed_in Y (subspace_topology X Tx Y) A.
We will prove ∃C : set, closed_in X Tx C A = C Y.
We prove the intermediate claim HTsubspace: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HY).
We prove the intermediate claim HAdef: topology_on Y (subspace_topology X Tx Y) (A Y ∃Usubspace_topology X Tx Y, A = Y U).
An exact proof term for the current goal is HAclosed.
We prove the intermediate claim HAandEx: A Y ∃Usubspace_topology X Tx Y, A = Y U.
An exact proof term for the current goal is (andER (topology_on Y (subspace_topology X Tx Y)) (A Y ∃Usubspace_topology X Tx Y, A = Y U) HAdef).
We prove the intermediate claim HexU: ∃Usubspace_topology X Tx Y, A = Y U.
An exact proof term for the current goal is (andER (A Y) (∃Usubspace_topology X Tx Y, A = Y U) HAandEx).
Apply HexU to the current goal.
Let U be given.
Assume HandEq.
Apply HandEq to the current goal.
Assume HUsubspace: U subspace_topology X Tx Y.
Assume HAeq: A = Y U.
We prove the intermediate claim HUexV: ∃VTx, U = V Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set∃VTx, U0 = V Y) U HUsubspace).
Apply HUexV to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V Tx.
Assume HUeq: U = V Y.
Set C to be the term X V.
We prove the intermediate claim HCclosed: closed_in X Tx C.
We will prove topology_on X Tx (C X ∃WTx, C = X W).
Apply andI to the current goal.
An exact proof term for the current goal is HTx.
Apply andI to the current goal.
An exact proof term for the current goal is (setminus_Subq X V).
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV.
Use reflexivity.
We prove the intermediate claim HAeqC: A = C Y.
rewrite the current goal using HAeq (from left to right).
rewrite the current goal using HUeq (from left to right).
We will prove Y (V Y) = (X V) Y.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x Y (V Y).
We prove the intermediate claim HxY: x Y.
An exact proof term for the current goal is (setminusE1 Y (V Y) x Hx).
We prove the intermediate claim HxnotVY: x V Y.
An exact proof term for the current goal is (setminusE2 Y (V Y) x Hx).
We prove the intermediate claim HxnotV: x V.
Assume HxV: x V.
Apply HxnotVY to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV.
An exact proof term for the current goal is HxY.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (HY x HxY).
Apply binintersectI to the current goal.
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
An exact proof term for the current goal is HxnotV.
An exact proof term for the current goal is HxY.
Let x be given.
Assume Hx: x (X V) Y.
We prove the intermediate claim HxXV: x X V.
An exact proof term for the current goal is (binintersectE1 (X V) Y x Hx).
We prove the intermediate claim HxY: x Y.
An exact proof term for the current goal is (binintersectE2 (X V) Y x Hx).
We prove the intermediate claim HxnotV: x V.
An exact proof term for the current goal is (setminusE2 X V x HxXV).
Apply setminusI to the current goal.
An exact proof term for the current goal is HxY.
Assume HxVY: x V Y.
Apply HxnotV to the current goal.
An exact proof term for the current goal is (binintersectE1 V Y x HxVY).
We use C to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HCclosed.
An exact proof term for the current goal is HAeqC.
Assume Hexists: ∃C : set, closed_in X Tx C A = C Y.
We will prove closed_in Y (subspace_topology X Tx Y) A.
Apply Hexists to the current goal.
Let C be given.
Assume HCandEq.
Apply HCandEq to the current goal.
Assume HCclosed: closed_in X Tx C.
Assume HAeq: A = C Y.
We prove the intermediate claim HTsubspace: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HY).
We prove the intermediate claim HCdef: topology_on X Tx (C X ∃VTx, C = X V).
An exact proof term for the current goal is HCclosed.
We prove the intermediate claim HCandEx: C X ∃VTx, C = X V.
An exact proof term for the current goal is (andER (topology_on X Tx) (C X ∃VTx, C = X V) HCdef).
We prove the intermediate claim HexV: ∃VTx, C = X V.
An exact proof term for the current goal is (andER (C X) (∃VTx, C = X V) HCandEx).
Apply HexV to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V Tx.
Assume HCeq: C = X V.
Set U to be the term V Y.
We prove the intermediate claim HUsubspace: U subspace_topology X Tx Y.
We prove the intermediate claim HUinPowerY: U 𝒫 Y.
Apply PowerI to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 V Y).
We prove the intermediate claim HPred: ∃WTx, U = W Y.
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV.
Use reflexivity.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set∃WTx, U0 = W Y) U HUinPowerY HPred).
We prove the intermediate claim HAeqYU: A = Y U.
rewrite the current goal using HAeq (from left to right).
rewrite the current goal using HCeq (from left to right).
We will prove (X V) Y = Y (V Y).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x (X V) Y.
We prove the intermediate claim HxXV: x X V.
An exact proof term for the current goal is (binintersectE1 (X V) Y x Hx).
We prove the intermediate claim HxY: x Y.
An exact proof term for the current goal is (binintersectE2 (X V) Y x Hx).
We prove the intermediate claim HxnotV: x V.
An exact proof term for the current goal is (setminusE2 X V x HxXV).
Apply setminusI to the current goal.
An exact proof term for the current goal is HxY.
Assume HxVY: x V Y.
Apply HxnotV to the current goal.
An exact proof term for the current goal is (binintersectE1 V Y x HxVY).
Let x be given.
Assume Hx: x Y (V Y).
We prove the intermediate claim HxY: x Y.
An exact proof term for the current goal is (setminusE1 Y (V Y) x Hx).
We prove the intermediate claim HxnotVY: x V Y.
An exact proof term for the current goal is (setminusE2 Y (V Y) x Hx).
We prove the intermediate claim HxnotV: x V.
Assume HxV: x V.
Apply HxnotVY to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV.
An exact proof term for the current goal is HxY.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (HY x HxY).
Apply binintersectI to the current goal.
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
An exact proof term for the current goal is HxnotV.
An exact proof term for the current goal is HxY.
We prove the intermediate claim HAsub: A Y.
rewrite the current goal using HAeq (from left to right).
An exact proof term for the current goal is (binintersect_Subq_2 C Y).
We will prove topology_on Y (subspace_topology X Tx Y) (A Y ∃Usubspace_topology X Tx Y, A = Y U).
Apply andI to the current goal.
An exact proof term for the current goal is HTsubspace.
Apply andI to the current goal.
An exact proof term for the current goal is HAsub.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUsubspace.
An exact proof term for the current goal is HAeqYU.
Theorem. (closed_in_closed_subspace)
∀X Tx Y A : set, topology_on X Txclosed_in X Tx Yclosed_in Y (subspace_topology X Tx Y) Aclosed_in X Tx A
Proof:
Let X, Tx, Y and A be given.
Assume HTx: topology_on X Tx.
Assume HY: closed_in X Tx Y.
Assume HA: closed_in Y (subspace_topology X Tx Y) A.
We will prove closed_in X Tx A.
We prove the intermediate claim HYsub: Y X.
An exact proof term for the current goal is (andEL (Y X) (∃UTx, Y = X U) (andER (topology_on X Tx) (Y X ∃UTx, Y = X U) HY)).
We prove the intermediate claim Hexists: ∃C : set, closed_in X Tx C A = C Y.
Apply (iffEL (closed_in Y (subspace_topology X Tx Y) A) (∃C : set, closed_in X Tx C A = C Y) (closed_in_subspace_iff_intersection X Tx Y A HTx HYsub)) to the current goal.
An exact proof term for the current goal is HA.
Apply Hexists to the current goal.
Let C be given.
Assume HCandA: closed_in X Tx C A = C Y.
We prove the intermediate claim HCclosed: closed_in X Tx C.
An exact proof term for the current goal is (andEL (closed_in X Tx C) (A = C Y) HCandA).
We prove the intermediate claim HAeq: A = C Y.
An exact proof term for the current goal is (andER (closed_in X Tx C) (A = C Y) HCandA).
We prove the intermediate claim HCexists: ∃UTx, C = X U.
An exact proof term for the current goal is (andER (C X) (∃UTx, C = X U) (andER (topology_on X Tx) (C X ∃UTx, C = X U) HCclosed)).
Apply HCexists to the current goal.
Let U be given.
Assume HU: U Tx C = X U.
We prove the intermediate claim HUinTx: U Tx.
An exact proof term for the current goal is (andEL (U Tx) (C = X U) HU).
We prove the intermediate claim HCeq: C = X U.
An exact proof term for the current goal is (andER (U Tx) (C = X U) HU).
We prove the intermediate claim HYexists: ∃VTx, Y = X V.
An exact proof term for the current goal is (andER (Y X) (∃VTx, Y = X V) (andER (topology_on X Tx) (Y X ∃VTx, Y = X V) HY)).
Apply HYexists to the current goal.
Let V be given.
Assume HV: V Tx Y = X V.
We prove the intermediate claim HVinTx: V Tx.
An exact proof term for the current goal is (andEL (V Tx) (Y = X V) HV).
We prove the intermediate claim HYeq: Y = X V.
An exact proof term for the current goal is (andER (V Tx) (Y = X V) HV).
We prove the intermediate claim HAeqSetminus: A = (X U) (X V).
rewrite the current goal using HAeq (from left to right).
rewrite the current goal using HCeq (from left to right).
rewrite the current goal using HYeq (from left to right).
Use reflexivity.
We prove the intermediate claim HDeM: (X U) (X V) = X (U V).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x (X U) (X V).
We will prove x X (U V).
We prove the intermediate claim HxXU: x X U.
An exact proof term for the current goal is (binintersectE1 (X U) (X V) x Hx).
We prove the intermediate claim HxXV: x X V.
An exact proof term for the current goal is (binintersectE2 (X U) (X V) x Hx).
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X U x HxXU).
We prove the intermediate claim HxninU: x U.
An exact proof term for the current goal is (setminusE2 X U x HxXU).
We prove the intermediate claim HxninV: x V.
An exact proof term for the current goal is (setminusE2 X V x HxXV).
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
Assume HxUV: x U V.
We will prove False.
Apply (binunionE U V x HxUV) to the current goal.
Assume HxU: x U.
An exact proof term for the current goal is (HxninU HxU).
Assume HxV: x V.
An exact proof term for the current goal is (HxninV HxV).
Let x be given.
Assume Hx: x X (U V).
We will prove x (X U) (X V).
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X (U V) x Hx).
We prove the intermediate claim HxninUV: x U V.
An exact proof term for the current goal is (setminusE2 X (U V) x Hx).
Apply binintersectI to the current goal.
We will prove x X U.
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
Assume HxU: x U.
We will prove False.
We prove the intermediate claim HxUV: x U V.
An exact proof term for the current goal is (binunionI1 U V x HxU).
An exact proof term for the current goal is (HxninUV HxUV).
We will prove x X V.
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
Assume HxV: x V.
We will prove False.
We prove the intermediate claim HxUV: x U V.
An exact proof term for the current goal is (binunionI2 U V x HxV).
An exact proof term for the current goal is (HxninUV HxUV).
We prove the intermediate claim HUV: U V Tx.
We prove the intermediate claim HUV_eq: U V = (UPair U V).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x U V.
We will prove x (UPair U V).
Apply (binunionE U V x Hx) to the current goal.
Assume HxU: x U.
Apply (UnionI (UPair U V) x U HxU) to the current goal.
An exact proof term for the current goal is (UPairI1 U V).
Assume HxV: x V.
Apply (UnionI (UPair U V) x V HxV) to the current goal.
An exact proof term for the current goal is (UPairI2 U V).
Let x be given.
Assume Hx: x (UPair U V).
We will prove x U V.
Apply (UnionE_impred (UPair U V) x Hx (x U V)) to the current goal.
Let W be given.
Assume HxW: x W.
Assume HWin: W UPair U V.
Apply (UPairE W U V HWin) to the current goal.
Assume HWeqU: W = U.
We prove the intermediate claim HxU: x U.
rewrite the current goal using HWeqU (from right to left).
An exact proof term for the current goal is HxW.
An exact proof term for the current goal is (binunionI1 U V x HxU).
Assume HWeqV: W = V.
We prove the intermediate claim HxV: x V.
rewrite the current goal using HWeqV (from right to left).
An exact proof term for the current goal is HxW.
An exact proof term for the current goal is (binunionI2 U V x HxV).
rewrite the current goal using HUV_eq (from left to right).
We prove the intermediate claim HUPairSub: UPair U V Tx.
Let W be given.
Assume HW: W UPair U V.
We will prove W Tx.
Apply (UPairE W U V HW) to the current goal.
Assume HWeqU: W = U.
rewrite the current goal using HWeqU (from left to right).
An exact proof term for the current goal is HUinTx.
Assume HWeqV: W = V.
rewrite the current goal using HWeqV (from left to right).
An exact proof term for the current goal is HVinTx.
An exact proof term for the current goal is (topology_union_closed X Tx (UPair U V) HTx HUPairSub).
We prove the intermediate claim HAeqFinal: A = X (U V).
rewrite the current goal using HAeqSetminus (from left to right).
An exact proof term for the current goal is HDeM.
We will prove topology_on X Tx (A X ∃U0Tx, A = X U0).
Apply andI to the current goal.
An exact proof term for the current goal is HTx.
Apply andI to the current goal.
We will prove A X.
rewrite the current goal using HAeqFinal (from left to right).
An exact proof term for the current goal is (setminus_Subq X (U V)).
We will prove ∃U0Tx, A = X U0.
We use (U V) to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUV.
An exact proof term for the current goal is HAeqFinal.
Theorem. (closure_in_subspace)
∀X Tx Y A : set, topology_on X TxY Xclosure_of Y (subspace_topology X Tx Y) A = (closure_of X Tx A) Y
Proof:
Let X, Tx, Y and A be given.
Assume HTx: topology_on X Tx.
Assume HY: Y X.
We will prove closure_of Y (subspace_topology X Tx Y) A = (closure_of X Tx A) Y.
We prove the intermediate claim HTy: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HY).
Apply set_ext to the current goal.
Let y be given.
Assume Hy: y closure_of Y (subspace_topology X Tx Y) A.
We will prove y (closure_of X Tx A) Y.
We prove the intermediate claim HyY: y Y.
An exact proof term for the current goal is (SepE1 Y (λy0 ⇒ ∀U : set, U subspace_topology X Tx Yy0 UU A Empty) y Hy).
We prove the intermediate claim HysubCond: ∀U : set, U subspace_topology X Tx Yy UU A Empty.
An exact proof term for the current goal is (SepE2 Y (λy0 ⇒ ∀U : set, U subspace_topology X Tx Yy0 UU A Empty) y Hy).
Apply binintersectI to the current goal.
We will prove y closure_of X Tx A.
We prove the intermediate claim HyX: y X.
An exact proof term for the current goal is (HY y HyY).
We prove the intermediate claim HyCond: ∀V : set, V Txy VV A Empty.
Let V be given.
Assume HV: V Tx.
Assume HyV: y V.
We will prove V A Empty.
Set U to be the term V Y.
We prove the intermediate claim HU: U subspace_topology X Tx Y.
We prove the intermediate claim HUinPower: U 𝒫 Y.
Apply PowerI to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 V Y).
We prove the intermediate claim HPred: ∃WTx, U = W Y.
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV.
Use reflexivity.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 ⇒ ∃WTx, U0 = W Y) U HUinPower HPred).
We prove the intermediate claim HyU: y U.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HyV.
An exact proof term for the current goal is HyY.
We prove the intermediate claim HUA_ne: U A Empty.
An exact proof term for the current goal is (HysubCond U HU HyU).
We will prove V A Empty.
Assume HVA_empty: V A = Empty.
We prove the intermediate claim HUA_sub_VA: U A V A.
Let z be given.
Assume Hz: z U A.
We prove the intermediate claim HzU: z U.
An exact proof term for the current goal is (binintersectE1 U A z Hz).
We prove the intermediate claim HzV: z V.
An exact proof term for the current goal is (binintersectE1 V Y z HzU).
We prove the intermediate claim HzA: z A.
An exact proof term for the current goal is (binintersectE2 U A z Hz).
An exact proof term for the current goal is (binintersectI V A z HzV HzA).
We prove the intermediate claim HUA_sub_Empty: U A Empty.
rewrite the current goal using HVA_empty (from right to left).
An exact proof term for the current goal is HUA_sub_VA.
We prove the intermediate claim HUA_empty: U A = Empty.
An exact proof term for the current goal is (Empty_Subq_eq (U A) HUA_sub_Empty).
An exact proof term for the current goal is (HUA_ne HUA_empty).
An exact proof term for the current goal is (SepI X (λy0 ⇒ ∀V : set, V Txy0 VV A Empty) y HyX HyCond).
An exact proof term for the current goal is HyY.
Let y be given.
Assume Hy: y (closure_of X Tx A) Y.
We will prove y closure_of Y (subspace_topology X Tx Y) A.
We prove the intermediate claim HyClX: y closure_of X Tx A.
An exact proof term for the current goal is (binintersectE1 (closure_of X Tx A) Y y Hy).
We prove the intermediate claim HyY: y Y.
An exact proof term for the current goal is (binintersectE2 (closure_of X Tx A) Y y Hy).
We prove the intermediate claim HyXCond: ∀V : set, V Txy VV A Empty.
An exact proof term for the current goal is (SepE2 X (λy0 ⇒ ∀V : set, V Txy0 VV A Empty) y HyClX).
We prove the intermediate claim HySubCond: ∀U : set, U subspace_topology X Tx Yy UU A Empty.
Let U be given.
Assume HU: U subspace_topology X Tx Y.
Assume HyU: y U.
We will prove U A Empty.
We prove the intermediate claim HUex: ∃VTx, U = V Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 ⇒ ∃VTx, U0 = V Y) U HU).
Apply HUex to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V Tx.
Assume HUeq: U = V Y.
We prove the intermediate claim HyV: y V.
We prove the intermediate claim HyVY: y V Y.
rewrite the current goal using HUeq (from right to left).
An exact proof term for the current goal is HyU.
An exact proof term for the current goal is (binintersectE1 V Y y HyVY).
We prove the intermediate claim HVA_ne: V A Empty.
An exact proof term for the current goal is (HyXCond V HV HyV).
rewrite the current goal using HUeq (from left to right).
We will prove (V Y) A Empty.
Assume HVYAempty: (V Y) A = Empty.
The rest of this subproof is missing.
An exact proof term for the current goal is (SepI Y (λy0 ⇒ ∀U : set, U subspace_topology X Tx Yy0 UU A Empty) y HyY HySubCond).
Theorem. (closure_characterization)
∀X Tx A x : set, topology_on X Tx(x closure_of X Tx A (∀UTx, x UU A Empty))
Proof:
Let X, Tx, A and x be given.
Assume HTx: topology_on X Tx.
We will prove x closure_of X Tx A (∀UTx, x UU A Empty).
Apply iffI to the current goal.
Assume Hx: x closure_of X Tx A.
We will prove ∀UTx, x UU A Empty.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∀U : set, U Txx0 UU A Empty) x Hx).
Assume Hcond: ∀UTx, x UU A Empty.
We will prove x closure_of X Tx A.
Apply xm (x X) to the current goal.
Assume HxX: x X.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U Txx0 UU A Empty) x HxX Hcond).
Assume HxnotX: x X.
Apply FalseE to the current goal.
The rest of this subproof is missing.
Definition. We define limit_point_of to be λX Tx A x ⇒ topology_on X Tx x X ∀U : set, U Txx U∃y : set, y A y x y U of type setsetsetsetprop.
Definition. We define limit_points_of to be λX Tx A ⇒ {xX|limit_point_of X Tx A x} of type setsetsetset.
Proof:
Let X, Tx and A be given.
Assume HTx: topology_on X Tx.
We will prove closure_of X Tx A = A limit_points_of X Tx A.
The rest of this subproof is missing.
Proof:
Let X, Tx and A be given.
Assume HTx: topology_on X Tx.
We will prove closed_in X Tx A limit_points_of X Tx A A.
The rest of this subproof is missing.
Definition. We define Hausdorff_space to be λX Tx ⇒ topology_on X Tx ∀x1 x2 : set, x1 x2∃U V : set, U Tx V Tx x1 U x2 V U V = Empty of type setsetprop.
Definition. We define T1_space to be λX Tx ⇒ topology_on X Tx (∀F : set, finite Fclosed_in X Tx F) of type setsetprop.
Theorem. (finite_sets_closed_in_Hausdorff)
∀X Tx : set, Hausdorff_space X Tx∀F : set, finite Fclosed_in X Tx F
Proof:
Let X and Tx be given.
Assume HH: Hausdorff_space X Tx.
Let F be given.
Assume HF: finite F.
We will prove closed_in X Tx F.
The rest of this subproof is missing.
Theorem. (limit_points_infinite_neighborhoods)
∀X Tx A x : set, T1_space X Tx(limit_point_of X Tx A x (∀UTx, x Uinfinite (U A)))
Proof:
Let X, Tx, A and x be given.
Assume HT1: T1_space X Tx.
We will prove limit_point_of X Tx A x (∀UTx, x Uinfinite (U A)).
The rest of this subproof is missing.
Theorem. (Hausdorff_unique_limits)
∀X Tx seq x y : set, Hausdorff_space X Txx yfunction_on seq ω X(∀U : set, U Txx U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U)(∀U : set, U Txy U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U)False
Proof:
Let X, Tx, seq, x and y be given.
Assume HH: Hausdorff_space X Tx.
Assume Hneq: x y.
Assume Hseq: function_on seq ω X.
Assume Hx: ∀U : set, U Txx U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U.
Assume Hy: ∀U : set, U Txy U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U.
We will prove False.
The rest of this subproof is missing.
Proof:
Let X, Tx, Y and Ty be given.
Assume H: Hausdorff_space X Tx Hausdorff_space Y Ty.
We will prove Hausdorff_space (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
Theorem. (ex17_1_topology_from_closed_sets)
∀X Tx : set, closed_in X Tx X(∀A : set, closed_in X Tx Aclosed_in X Tx (X A))topology_on X Tx
Proof:
Let X and Tx be given.
Assume H1: closed_in X Tx X.
Assume H2: ∀A : set, closed_in X Tx Aclosed_in X Tx (X A).
We will prove topology_on X Tx.
The rest of this subproof is missing.
Theorem. (ex17_2_closed_in_closed_subspace)
∀X Tx Y A : set, closed_in X Tx Yclosed_in Y (subspace_topology X Tx Y) Aclosed_in X Tx A
Proof:
Let X, Tx, Y and A be given.
Assume HY: closed_in X Tx Y.
Assume HA: closed_in Y (subspace_topology X Tx Y) A.
We will prove closed_in X Tx A.
We prove the intermediate claim HTx: topology_on X Tx.
An exact proof term for the current goal is (andEL (topology_on X Tx) (Y X ∃UTx, Y = X U) HY).
An exact proof term for the current goal is (closed_in_closed_subspace X Tx Y A HTx HY HA).
Theorem. (ex17_3_product_of_closed_sets_closed)
∀X Tx Y Ty A B : set, closed_in X Tx Aclosed_in Y Ty Bclosed_in (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B)
Proof:
Let X, Tx, Y, Ty, A and B be given.
Assume HA: closed_in X Tx A.
Assume HB: closed_in Y Ty B.
We will prove closed_in (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B).
The rest of this subproof is missing.
Theorem. (ex17_4_open_minus_closed_and_closed_minus_open)
∀X Tx U A : set, topology_on X Txopen_in X Tx Uclosed_in X Tx Aopen_in X Tx (U A) closed_in X Tx (A U)
Proof:
Let X, Tx, U and A be given.
Assume Htop: topology_on X Tx.
Assume HU: open_in X Tx U.
Assume HA: closed_in X Tx A.
We will prove open_in X Tx (U A) closed_in X Tx (A U).
We prove the intermediate claim HUtop: U Tx.
An exact proof term for the current goal is (andER (topology_on X Tx) (U Tx) HU).
We prove the intermediate claim HAdef: A X (∃VTx, A = X V).
An exact proof term for the current goal is (andER (topology_on X Tx) (A X (∃VTx, A = X V)) HA).
We prove the intermediate claim HexV: ∃VTx, A = X V.
An exact proof term for the current goal is (andER (A X) (∃VTx, A = X V) HAdef).
Apply HexV to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V Tx.
Assume HAeq: A = X V.
Apply andI to the current goal.
We will prove open_in X Tx (U A).
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (topology_elem_subset X Tx U Htop HUtop).
We prove the intermediate claim HUminusA_eq_UinterV: U A = U V.
rewrite the current goal using HAeq (from left to right).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x U (X V).
We prove the intermediate claim HxU: x U.
An exact proof term for the current goal is (setminusE1 U (X V) x Hx).
We prove the intermediate claim HxnotXV: x X V.
An exact proof term for the current goal is (setminusE2 U (X V) x Hx).
We prove the intermediate claim HxV: x V.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (HUsub x HxU).
Apply xm (x V) to the current goal.
Assume Hv.
An exact proof term for the current goal is Hv.
Assume Hnv.
Apply FalseE to the current goal.
Apply HxnotXV to the current goal.
An exact proof term for the current goal is (setminusI X V x HxX Hnv).
An exact proof term for the current goal is (binintersectI U V x HxU HxV).
Let x be given.
Assume Hx: x U V.
We prove the intermediate claim HxU: x U.
An exact proof term for the current goal is (binintersectE1 U V x Hx).
We prove the intermediate claim HxV: x V.
An exact proof term for the current goal is (binintersectE2 U V x Hx).
We prove the intermediate claim HxnotXV: x X V.
Assume H.
Apply (setminusE2 X V x H) to the current goal.
An exact proof term for the current goal is HxV.
An exact proof term for the current goal is (setminusI U (X V) x HxU HxnotXV).
rewrite the current goal using HUminusA_eq_UinterV (from left to right).
We prove the intermediate claim HUinterV: U V Tx.
An exact proof term for the current goal is (topology_binintersect_closed X Tx U V Htop HUtop HV).
An exact proof term for the current goal is (andI (topology_on X Tx) (U V Tx) Htop HUinterV).
We will prove closed_in X Tx (A U).
We prove the intermediate claim HAsub: A X.
An exact proof term for the current goal is (andEL (A X) (∃V0Tx, A = X V0) HAdef).
We prove the intermediate claim HAminusU_sub: A U X.
Let x be given.
Assume Hx.
We prove the intermediate claim HxA: x A.
An exact proof term for the current goal is (setminusE1 A U x Hx).
An exact proof term for the current goal is (HAsub x HxA).
We prove the intermediate claim HVU: V U Tx.
Set UFam to be the term UPair V U.
We prove the intermediate claim HUFamsub: UFam Tx.
Let W be given.
Assume HW: W UFam.
Apply (UPairE W V U HW) to the current goal.
Assume HWeqV.
rewrite the current goal using HWeqV (from left to right).
An exact proof term for the current goal is HV.
Assume HWeqU.
rewrite the current goal using HWeqU (from left to right).
An exact proof term for the current goal is HUtop.
We prove the intermediate claim HUnionVU: UFam = V U.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x UFam.
Apply UnionE_impred UFam x Hx to the current goal.
Let W be given.
Assume HxW: x W.
Assume HW: W UFam.
Apply (UPairE W V U HW) to the current goal.
Assume HWeqV.
We prove the intermediate claim HxV: x V.
rewrite the current goal using HWeqV (from right to left).
An exact proof term for the current goal is HxW.
An exact proof term for the current goal is (binunionI1 V U x HxV).
Assume HWeqU.
We prove the intermediate claim HxU: x U.
rewrite the current goal using HWeqU (from right to left).
An exact proof term for the current goal is HxW.
An exact proof term for the current goal is (binunionI2 V U x HxU).
Let x be given.
Assume Hx: x V U.
Apply (binunionE V U x Hx) to the current goal.
Assume HxV.
An exact proof term for the current goal is (UnionI UFam x V HxV (UPairI1 V U)).
Assume HxU.
An exact proof term for the current goal is (UnionI UFam x U HxU (UPairI2 V U)).
rewrite the current goal using HUnionVU (from right to left).
An exact proof term for the current goal is (topology_union_closed X Tx UFam Htop HUFamsub).
We prove the intermediate claim HAminusU_eq_XminusVU: A U = X (V U).
rewrite the current goal using HAeq (from left to right).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x (X V) U.
We prove the intermediate claim HxXV: x X V.
An exact proof term for the current goal is (setminusE1 (X V) U x Hx).
We prove the intermediate claim HxnotU: x U.
An exact proof term for the current goal is (setminusE2 (X V) U x Hx).
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X V x HxXV).
We prove the intermediate claim HxnotV: x V.
An exact proof term for the current goal is (setminusE2 X V x HxXV).
We prove the intermediate claim HxnotVU: x V U.
Assume H.
Apply (binunionE V U x H) to the current goal.
Assume HxV.
An exact proof term for the current goal is (HxnotV HxV).
Assume HxU.
An exact proof term for the current goal is (HxnotU HxU).
An exact proof term for the current goal is (setminusI X (V U) x HxX HxnotVU).
Let x be given.
Assume Hx: x X (V U).
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X (V U) x Hx).
We prove the intermediate claim HxnotVU: x V U.
An exact proof term for the current goal is (setminusE2 X (V U) x Hx).
We prove the intermediate claim HxnotV: x V.
Assume HxV.
Apply HxnotVU to the current goal.
An exact proof term for the current goal is (binunionI1 V U x HxV).
We prove the intermediate claim HxnotU: x U.
Assume HxU.
Apply HxnotVU to the current goal.
An exact proof term for the current goal is (binunionI2 V U x HxU).
We prove the intermediate claim HxXV: x X V.
An exact proof term for the current goal is (setminusI X V x HxX HxnotV).
An exact proof term for the current goal is (setminusI (X V) U x HxXV HxnotU).
We prove the intermediate claim HPred: ∃WTx, A U = X W.
We use (V U) to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HVU.
An exact proof term for the current goal is HAminusU_eq_XminusVU.
An exact proof term for the current goal is (andI (topology_on X Tx) (A U X (∃WTx, A U = X W)) Htop (andI (A U X) (∃WTx, A U = X W) HAminusU_sub HPred)).
Proof:
Let X be given.
The rest of this subproof is missing.
Theorem. (ex17_6_closure_properties)
∀X Tx A : set, topology_on X Txclosure_of X Tx (closure_of X Tx A) = closure_of X Tx A closed_in X Tx (closure_of X Tx A)
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
We will prove closure_of X Tx (closure_of X Tx A) = closure_of X Tx A closed_in X Tx (closure_of X Tx A).
The rest of this subproof is missing.
Theorem. (ex17_7_counterexample_union_closure)
∀X Tx A B : set, topology_on X Txclosed_in X Tx (A B)¬ (closed_in X Tx A closed_in X Tx B)
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HAB: closed_in X Tx (A B).
We will prove ¬ (closed_in X Tx A closed_in X Tx B).
The rest of this subproof is missing.
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
We will prove closure_of X Tx (A B) closure_of X Tx A closure_of X Tx B.
Let x be given.
Assume Hx: x closure_of X Tx (A B).
We will prove x closure_of X Tx A closure_of X Tx B.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (SepE1 X (λx0 ⇒ ∀U : set, U Txx0 UU (A B) Empty) x Hx).
We prove the intermediate claim HxAB: ∀U : set, U Txx UU (A B) Empty.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∀U : set, U Txx0 UU (A B) Empty) x Hx).
Apply binintersectI to the current goal.
We will prove x closure_of X Tx A.
We prove the intermediate claim HxA: x X (∀U : set, U Txx UU A Empty).
Apply andI to the current goal.
An exact proof term for the current goal is HxX.
Let U be given.
Assume HU: U Tx.
Assume HxU: x U.
We will prove U A Empty.
We prove the intermediate claim HABne: U (A B) Empty.
An exact proof term for the current goal is (HxAB U HU HxU).
Assume Hempty: U A = Empty.
Apply HABne to the current goal.
Apply Empty_Subq_eq to the current goal.
Let y be given.
Assume Hy: y U (A B).
We prove the intermediate claim HyU: y U.
An exact proof term for the current goal is (binintersectE1 U (A B) y Hy).
We prove the intermediate claim HyAB: y A B.
An exact proof term for the current goal is (binintersectE2 U (A B) y Hy).
We prove the intermediate claim HyA: y A.
An exact proof term for the current goal is (binintersectE1 A B y HyAB).
We prove the intermediate claim HyUA: y U A.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HyU.
An exact proof term for the current goal is HyA.
rewrite the current goal using Hempty (from right to left).
An exact proof term for the current goal is HyUA.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U Txx0 UU A Empty) x HxX (andER (x X) (∀U : set, U Txx UU A Empty) HxA)).
We will prove x closure_of X Tx B.
We prove the intermediate claim HxB: x X (∀U : set, U Txx UU B Empty).
Apply andI to the current goal.
An exact proof term for the current goal is HxX.
Let U be given.
Assume HU: U Tx.
Assume HxU: x U.
We will prove U B Empty.
We prove the intermediate claim HABne: U (A B) Empty.
An exact proof term for the current goal is (HxAB U HU HxU).
Assume Hempty: U B = Empty.
Apply HABne to the current goal.
Apply Empty_Subq_eq to the current goal.
Let y be given.
Assume Hy: y U (A B).
We prove the intermediate claim HyU: y U.
An exact proof term for the current goal is (binintersectE1 U (A B) y Hy).
We prove the intermediate claim HyAB: y A B.
An exact proof term for the current goal is (binintersectE2 U (A B) y Hy).
We prove the intermediate claim HyB: y B.
An exact proof term for the current goal is (binintersectE2 A B y HyAB).
We prove the intermediate claim HyUB: y U B.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HyU.
An exact proof term for the current goal is HyB.
rewrite the current goal using Hempty (from right to left).
An exact proof term for the current goal is HyUB.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U Txx0 UU B Empty) x HxX (andER (x X) (∀U : set, U Txx UU B Empty) HxB)).
Proof:
Let X, Y, Tx, Ty, A and B be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove closure_of (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B) = OrderedPair (closure_of X Tx A) (closure_of Y Ty B).
The rest of this subproof is missing.
Proof:
Let X be given.
We will prove Hausdorff_space X (order_topology X).
The rest of this subproof is missing.
Proof:
Let X, Tx, Y and Ty be given.
Assume HX: Hausdorff_space X Tx.
Assume HY: Hausdorff_space Y Ty.
We will prove Hausdorff_space (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
Proof:
Let X, Tx and Y be given.
Assume HX: Hausdorff_space X Tx.
We will prove Hausdorff_space Y (subspace_topology X Tx Y).
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
We will prove Hausdorff_space X Tx closed_in (OrderedPair X X) (product_topology X Tx X Tx) {OrderedPair x x|xX}.
The rest of this subproof is missing.
Theorem. (ex17_14_sequence_in_finite_complement_topology)
∀X seq : set, function_on seq ω X∀x : set, x X∀U : set, U finite_complement_topology Xx U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U
Proof:
Let X and seq be given.
Assume Hseq: function_on seq ω X.
Let x be given.
Assume Hx: x X.
Let U be given.
Assume HxU: x U.
We will prove ∃N : set, N ω ∀n : set, n ωN napply_fun seq n U.
The rest of this subproof is missing.
Theorem. (ex17_15_T1_characterization)
∀X Tx : set, topology_on X Tx(T1_space X Tx (∀x : set, closed_in X Tx {x}))
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
We will prove T1_space X Tx (∀x : set, closed_in X Tx {x}).
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define boundary_of to be λX Tx A ⇒ closure_of X Tx A closure_of X Tx (X A) of type setsetsetset.
Theorem. (ex17_19_boundary_properties)
∀X Tx A : set, topology_on X Txboundary_of X Tx A closure_of X Tx A boundary_of X Tx A closure_of X Tx (X A)
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
We will prove boundary_of X Tx A closure_of X Tx A boundary_of X Tx A closure_of X Tx (X A).
Apply andI to the current goal.
We will prove boundary_of X Tx A closure_of X Tx A.
An exact proof term for the current goal is (binintersect_Subq_1 (closure_of X Tx A) (closure_of X Tx (X A))).
We will prove boundary_of X Tx A closure_of X Tx (X A).
An exact proof term for the current goal is (binintersect_Subq_2 (closure_of X Tx A) (closure_of X Tx (X A))).
Proof:
Let X be given.
We will prove closure_of X (discrete_topology X) (X Empty) = X.
The rest of this subproof is missing.
Definition. We define preimage_of to be λX f V ⇒ {xX|apply_fun f x V} of type setsetsetset.
Definition. We define continuous_map to be λX Tx Y Ty f ⇒ topology_on X Tx topology_on Y Ty function_on f X Y ∀V : set, V Typreimage_of X f V Tx of type setsetsetsetsetprop.
Definition. We define continuous_at to be λf x ⇒ function_on f R R x R ∀eps : set, eps R∃delta : set, delta R True of type setsetprop.
Theorem. (continuity_equiv_forms)
∀X Tx Y Ty f : set, topology_on X Txtopology_on Y Ty(continuous_map X Tx Y Ty f (∀V : set, V Typreimage_of X f V Tx) (∀C : set, closed_in Y Ty Cclosed_in X Tx (preimage_of X f C)) (∀x : set, x X∀V : set, V Tyapply_fun f x V∃U : set, U Tx x U ∀u : set, u Uapply_fun f u V))
Proof:
Let X, Tx, Y, Ty and f be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove continuous_map X Tx Y Ty f (∀V : set, V Typreimage_of X f V Tx) (∀C : set, closed_in Y Ty Cclosed_in X Tx (preimage_of X f C)) (∀x : set, x X∀V : set, V Tyapply_fun f x V∃U : set, U Tx x U ∀u : set, u Uapply_fun f u V).
The rest of this subproof is missing.
Theorem. (identity_continuous)
∀X Tx : set, topology_on X Txlet id ≔ {UPair x x|xX} in continuous_map X Tx X Tx id
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove let id ≔ {UPair x x|xX} in continuous_map X Tx X Tx id.
The rest of this subproof is missing.
Definition. We define compose_fun to be λX f g ⇒ {UPair x (apply_fun g (apply_fun f x))|xX} of type setsetsetset.
Theorem. (composition_continuous)
∀X Tx Y Ty Z Tz f g : set, continuous_map X Tx Y Ty fcontinuous_map Y Ty Z Tz gcontinuous_map X Tx Z Tz (compose_fun X f g)
Proof:
Let X, Tx, Y, Ty, Z, Tz, f and g be given.
Assume Hf: continuous_map X Tx Y Ty f.
Assume Hg: continuous_map Y Ty Z Tz g.
We will prove continuous_map X Tx Z Tz (compose_fun X f g).
The rest of this subproof is missing.
Theorem. (continuous_construction_rules)
∀X Tx Y Ty Z Tz f g : set, continuous_map X Tx Y Ty fcontinuous_map X Tx Y Ty gcontinuous_map X Tx Y Ty f continuous_map X Tx Y Ty g continuous_map X Tx Y Ty g
Proof:
Let X, Tx, Y, Ty, Z, Tz, f and g be given.
Assume Hf: continuous_map X Tx Y Ty f.
Assume Hg: continuous_map X Tx Y Ty g.
We will prove continuous_map X Tx Y Ty f continuous_map X Tx Y Ty g continuous_map X Tx Y Ty g.
An exact proof term for the current goal is (andI (continuous_map X Tx Y Ty f continuous_map X Tx Y Ty g) (continuous_map X Tx Y Ty g) (andI (continuous_map X Tx Y Ty f) (continuous_map X Tx Y Ty g) Hf Hg) Hg).
Definition. We define homeomorphism to be λX Tx Y Ty f ⇒ continuous_map X Tx Y Ty f ∃g : set, continuous_map Y Ty X Tx g (∀x : set, x Xapply_fun g (apply_fun f x) = x) (∀y : set, y Yapply_fun f (apply_fun g y) = y) of type setsetsetsetsetprop.
Theorem. (continuous_on_subspace)
∀X Tx Y Ty f A : set, topology_on X TxA Xcontinuous_map X Tx Y Ty fcontinuous_map A (subspace_topology X Tx A) Y Ty f
Proof:
Let X, Tx, Y, Ty, f and A be given.
Assume HTx: topology_on X Tx.
Assume HA: A X.
Assume Hf: continuous_map X Tx Y Ty f.
We will prove continuous_map A (subspace_topology X Tx A) Y Ty f.
The rest of this subproof is missing.
Theorem. (homeomorphism_inverse_continuous)
∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty fcontinuous_map Y Ty X Tx f
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hhom: homeomorphism X Tx Y Ty f.
We will prove continuous_map Y Ty X Tx f.
The rest of this subproof is missing.
Theorem. (pasting_lemma)
∀X A B Y Tx Ty f g : set, topology_on X TxA TxB TxA B = Emptycontinuous_map A (subspace_topology X Tx A) Y Ty fcontinuous_map B (subspace_topology X Tx B) Y Ty gcontinuous_map (A B) (subspace_topology X Tx (A B)) Y Ty (f g)
Proof:
Let X, A, B, Y, Tx, Ty, f and g be given.
Assume HTx: topology_on X Tx.
Assume HA: A Tx.
Assume HB: B Tx.
Assume Hdisj: A B = Empty.
Assume Hf: continuous_map A (subspace_topology X Tx A) Y Ty f.
Assume Hg: continuous_map B (subspace_topology X Tx B) Y Ty g.
We will prove continuous_map (A B) (subspace_topology X Tx (A B)) Y Ty (f g).
The rest of this subproof is missing.
Theorem. (maps_into_products)
∀A X Tx Y Ty f g : set, continuous_map A Tx X Ty fcontinuous_map A Tx Y Ty gcontinuous_map A Tx (OrderedPair X Y) (product_topology X Ty Y Ty) (f g)
Proof:
Let A, X, Tx, Y, Ty, f and g be given.
Assume Hf: continuous_map A Tx X Ty f.
Assume Hg: continuous_map A Tx Y Ty g.
We will prove continuous_map A Tx (OrderedPair X Y) (product_topology X Ty Y Ty) (f g).
The rest of this subproof is missing.
Definition. We define projection_map to be λX Y ⇒ projection1 X Y of type setsetset.
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove continuous_map (OrderedPair X Y) (product_topology X Tx Y Ty) X Tx (projection_map X Y) continuous_map (OrderedPair X Y) (product_topology X Tx Y Ty) Y Ty (projection_map Y X).
The rest of this subproof is missing.
Theorem. (product_topology_universal)
∀X Tx Y Ty : set, topology_on X Txtopology_on Y Ty∃Tprod : set, topology_on (OrderedPair X Y) Tprod continuous_map (OrderedPair X Y) Tprod X Tx (projection_map X Y) continuous_map (OrderedPair X Y) Tprod Y Ty (projection_map Y X)
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove ∃Tprod : set, topology_on (OrderedPair X Y) Tprod continuous_map (OrderedPair X Y) Tprod X Tx (projection_map X Y) continuous_map (OrderedPair X Y) Tprod Y Ty (projection_map Y X).
The rest of this subproof is missing.
Definition. We define metric_on to be λX d ⇒ function_on d (OrderedPair X X) R (∀x y : set, x Xy Xapply_fun d (OrderedPair x y) = apply_fun d (OrderedPair y x)) (∀x : set, x Xapply_fun d (OrderedPair x x) = 0) (∀x y : set, x Xy X¬ (Rlt (apply_fun d (OrderedPair x y)) 0) apply_fun d (OrderedPair x y) = 0x = y) (∀x y z : set, x Xy Xz XRlt (apply_fun d (OrderedPair x z)) (apply_fun d (OrderedPair x y) apply_fun d (OrderedPair y z))) of type setsetprop.
Definition. We define open_ball to be λX d x ⇒ {yX|∃rR, Rlt (d x y) r} of type setsetsetset.
Definition. We define metric_topology to be λX d ⇒ generated_topology X {open_ball X d x|xX} of type setsetset.
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
We will prove basis_on X {open_ball X d x|xX}.
The rest of this subproof is missing.
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
We will prove topology_on X (metric_topology X d).
The rest of this subproof is missing.
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
The rest of this subproof is missing.
Theorem. (metric_epsilon_delta_continuity)
∀X dX Y dY f : set, metric_on X dXmetric_on Y dY(continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f (∀x0 : set, x0 X∀eps : set, eps R Rlt 0 eps∃delta : set, delta R Rlt 0 delta (∀x : set, x XRlt (apply_fun dX (OrderedPair x x0)) deltaRlt (apply_fun dY (OrderedPair (apply_fun f x) (apply_fun f x0))) eps)))
Proof:
Let X, dX, Y, dY and f be given.
Assume HdX: metric_on X dX.
Assume HdY: metric_on Y dY.
We will prove continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f (∀x0 : set, x0 X∀eps : set, eps R Rlt 0 eps∃delta : set, delta R Rlt 0 delta (∀x : set, x XRlt (apply_fun dX (OrderedPair x x0)) deltaRlt (apply_fun dY (OrderedPair (apply_fun f x) (apply_fun f x0))) eps)).
The rest of this subproof is missing.
Definition. We define sequence_in to be λseq A ⇒ seq A of type setsetprop.
Definition. We define sequence_on to be λseq A ⇒ function_on seq ω A of type setsetprop.
Definition. We define converges_to to be λX Tx seq x ⇒ topology_on X Tx sequence_on seq X x X ∀U : set, U Txx U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U of type setsetsetsetprop.
Definition. We define image_of to be λf seq ⇒ Repl seq (λy ⇒ y) of type setsetset.
Definition. We define function_sequence_value to be λf_seq n x ⇒ apply_fun (apply_fun f_seq n) x of type setsetsetset.
Definition. We define sequence_converges_metric to be λX d seq x ⇒ metric_on X d sequence_on seq X x X ∀eps : set, eps R Rlt 0 eps∃N : set, N ω ∀n : set, n ωN nRlt (apply_fun d (OrderedPair (apply_fun seq n) x)) eps of type setsetsetsetprop.
Theorem. (metric_limits_unique)
∀X d seq x y : set, metric_on X dsequence_on seq Xsequence_converges_metric X d seq xsequence_converges_metric X d seq yx = y
Proof:
Let X, d, seq, x and y be given.
Assume Hd: metric_on X d.
Assume Hseq: sequence_on seq X.
Assume Hx: sequence_converges_metric X d seq x.
Assume Hy: sequence_converges_metric X d seq y.
We will prove x = y.
The rest of this subproof is missing.
Definition. We define uniform_convergence_functions to be λX dX Y dY f_seq f ⇒ metric_on X dX metric_on Y dY function_on f_seq ω (function_space X Y) function_on f X Y (∀n : set, n ωfunction_on (apply_fun f_seq n) X Y) ∀eps : set, eps R Rlt 0 eps∃N : set, N ω ∀n : set, n ωN n∀x : set, x XRlt (apply_fun dY (OrderedPair (apply_fun (apply_fun f_seq n) x) (apply_fun f x))) eps of type setsetsetsetsetsetprop.
Theorem. (uniform_limit_of_continuous_is_continuous)
∀X dX Y dY f_seq f : set, metric_on X dXmetric_on Y dYfunction_on f_seq ω (function_space X Y)(∀n : set, n ωcontinuous_map X (metric_topology X dX) Y (metric_topology Y dY) (apply_fun f_seq n))uniform_convergence_functions X dX Y dY f_seq fcontinuous_map X (metric_topology X dX) Y (metric_topology Y dY) f
Proof:
Let X, dX, Y, dY, f_seq and f be given.
Assume HdX: metric_on X dX.
Assume HdY: metric_on Y dY.
Assume Hfseq: function_on f_seq ω (function_space X Y).
Assume Hcont: ∀n : set, n ωcontinuous_map X (metric_topology X dX) Y (metric_topology Y dY) (apply_fun f_seq n).
Assume Hunif: uniform_convergence_functions X dX Y dY f_seq f.
We will prove continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f.
The rest of this subproof is missing.
Proof:
Let X, d, seq and x be given.
Assume H: sequence_converges_metric X d seq x.
We will prove sequence_converges_metric X d seq x.
An exact proof term for the current goal is H.
Proof:
Let X, dX, Y, dY and f be given.
Assume HdX: metric_on X dX.
Assume HdY: metric_on Y dY.
We will prove continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f ∀seq x : set, sequence_converges_metric X dX seq xsequence_converges_metric Y dY ({OrderedPair n (apply_fun f (apply_fun seq n))|nω}) (apply_fun f x).
The rest of this subproof is missing.
Definition. We define quotient_topology to be λX Tx Y f ⇒ {V𝒫 Y|{xX|apply_fun f x V} Tx} of type setsetsetsetset.
Definition. We define quotient_map to be λX Tx Y f ⇒ topology_on X Tx function_on f X Y (∀y : set, y Y∃x : set, x X apply_fun f x = y) of type setsetsetsetprop.
Proof:
Let X, Tx, Y and f be given.
Assume HTx: topology_on X Tx.
Assume Hf: quotient_map X Tx Y f.
We will prove topology_on Y (quotient_topology X Tx Y f).
The rest of this subproof is missing.
Theorem. (quotient_universal_property)
∀X Tx Y Ty f : set, quotient_map X Tx Y ftopology_on Y Tycontinuous_map X Tx Y Ty f
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hf: quotient_map X Tx Y f.
Assume HTy: topology_on Y Ty.
We will prove continuous_map X Tx Y Ty f.
The rest of this subproof is missing.
Definition. We define separation_of to be λX U V ⇒ U 𝒫 X V 𝒫 X U V = Empty U Empty V Empty of type setsetsetprop.
Definition. We define connected_space to be λX Tx ⇒ topology_on X Tx ¬ (∃U V : set, U Tx V Tx separation_of X U V U V = X) of type setsetprop.
Proof:
Let X and Tx be given.
We will prove connected_space X Tx ¬ (∃A : set, A Empty A X open_in X Tx A closed_in X Tx A).
The rest of this subproof is missing.
Theorem. (separation_subspace_limit_points)
∀X Tx Y A B : set, topology_on X TxA B = EmptyA B = Yopen_in X Tx Aopen_in X Tx B∃a b : set, limit_point_of X Tx A a limit_point_of X Tx B b a Y b Y
Proof:
Let X, Tx, Y, A and B be given.
Assume HTx: topology_on X Tx.
Assume Hdisj: A B = Empty.
Assume Hunion: A B = Y.
Assume HA: open_in X Tx A.
Assume HB: open_in X Tx B.
We will prove ∃a b : set, limit_point_of X Tx A a limit_point_of X Tx B b a Y b Y.
The rest of this subproof is missing.
Theorem. (connected_subset_in_separation_side)
∀X Tx C D Y : set, topology_on X Txconnected_space Y TxC D = EmptyC D = Xopen_in X Tx Copen_in X Tx DY C Y D
Proof:
Let X, Tx, C, D and Y be given.
Assume HTx: topology_on X Tx.
Assume HY: connected_space Y Tx.
Assume Hdisj: C D = Empty.
Assume Hunion: C D = X.
Assume HC: open_in X Tx C.
Assume HD: open_in X Tx D.
We will prove Y C Y D.
The rest of this subproof is missing.
Theorem. (union_connected_common_point)
∀X Tx F : set, topology_on X Tx(∀C : set, C Fconnected_space C (subspace_topology X Tx C))(∃x : set, ∀C : set, C Fx C)connected_space ( F) (subspace_topology X Tx ( F))
Proof:
Let X, Tx and F be given.
Assume HTx: topology_on X Tx.
Assume HF: ∀C : set, C Fconnected_space C (subspace_topology X Tx C).
Assume Hcommon: ∃x : set, ∀C : set, C Fx C.
We will prove connected_space ( F) (subspace_topology X Tx ( F)).
The rest of this subproof is missing.
Proof:
Let X, Tx, A and b be given.
Assume HTx: topology_on X Tx.
Assume HA: connected_space A (subspace_topology X Tx A).
Assume Hb: limit_point_of X Tx A b.
We will prove connected_space (A {b}) (subspace_topology X Tx (A {b})).
The rest of this subproof is missing.
Theorem. (continuous_image_connected)
∀X Tx Y Ty f : set, connected_space X Txcontinuous_map X Tx Y Ty fconnected_space Y Ty
Proof:
Let X, Tx, Y, Ty and f be given.
Assume HX: connected_space X Tx.
Assume Hf: continuous_map X Tx Y Ty f.
We will prove connected_space Y Ty.
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (connected_subsets_real_are_intervals)
∀A : set, A Rconnected_space A (subspace_topology R R_standard_topology A)∀x y z : set, x Ay Az R(Rlt x z Rlt z y Rlt y z Rlt z x)z A
Proof:
Let A be given.
Assume HA: A R.
Let x, y and z be given.
Assume Hx: x A.
Assume Hy: y A.
Assume Hz: z R.
Assume Hbetw: Rlt x z Rlt z y Rlt y z Rlt z x.
We will prove z A.
The rest of this subproof is missing.
Proof:
Let X, Tx, Y and Ty be given.
Assume HX: connected_space X Tx.
Assume HY: connected_space Y Ty.
We will prove connected_space (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define path_between to be λX x y p ⇒ function_on p unit_interval X apply_fun p 0 = x apply_fun p 1 = y of type setsetsetsetprop.
Definition. We define path_connected_space to be λX Tx ⇒ topology_on X Tx ∀x y : set, x Xy X∃p : set, path_between X x y p of type setsetprop.
Proof:
Let X and Tx be given.
Assume Hpath: path_connected_space X Tx.
We will prove connected_space X Tx.
The rest of this subproof is missing.
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hpath: path_connected_space X Tx.
Assume Hf: continuous_map X Tx Y Ty f.
We will prove path_connected_space Y Ty.
The rest of this subproof is missing.
Definition. We define path_component_of to be λX Tx x ⇒ {yX|∃p : set, function_on p unit_interval X continuous_map unit_interval R_standard_topology X Tx p apply_fun p 0 = x apply_fun p 1 = y} of type setsetsetset.
Theorem. (path_components_equivalence_relation)
∀X Tx : set, topology_on X Tx(∀x : set, x Xx path_component_of X Tx x) (∀x y : set, x Xy Xy path_component_of X Tx xx path_component_of X Tx y) (∀x y z : set, x Xy Xz Xy path_component_of X Tx xz path_component_of X Tx yz path_component_of X Tx x)
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (∀x : set, x Xx path_component_of X Tx x) (∀x y : set, x Xy Xy path_component_of X Tx xx path_component_of X Tx y) (∀x y z : set, x Xy Xz Xy path_component_of X Tx xz path_component_of X Tx yz path_component_of X Tx x).
The rest of this subproof is missing.
Definition. We define component_of to be λX Tx x ⇒ {yX|∃C : set, connected_space C (subspace_topology X Tx C) x C y C} of type setsetsetset.
Definition. We define locally_connected to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∀U : set, U Txx U∃V : set, V Tx x V V U connected_space V (subspace_topology X Tx V) of type setsetprop.
Definition. We define locally_path_connected to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∃U : set, U Tx x U path_connected_space U (subspace_topology X Tx U) of type setsetprop.
Definition. We define pairwise_disjoint to be λFam ⇒ ∀U V : set, U FamV FamU VU V = Empty of type setprop.
Definition. We define covers to be λX U ⇒ ∀x : set, x X∃u : set, u U x u of type setsetprop.
Theorem. (path_components_open)
∀X Tx : set, locally_path_connected X Tx∀x : set, x Xopen_in X Tx (path_component_of X Tx x)
Proof:
Let X and Tx be given.
Assume Hlpc: locally_path_connected X Tx.
Let x be given.
Assume Hx: x X.
We will prove open_in X Tx (path_component_of X Tx x).
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hlpc: locally_path_connected X Tx.
Let x be given.
Assume Hx: x X.
We will prove path_component_of X Tx x = component_of X Tx x.
The rest of this subproof is missing.
Theorem. (components_are_closed)
∀X Tx : set, topology_on X Tx∀x : set, x Xclosed_in X Tx (component_of X Tx x)
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
Let x be given.
Assume Hx: x X.
We will prove closed_in X Tx (component_of X Tx x).
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
The rest of this subproof is missing.
Proof:
Let X, Tx, Y and f be given.
Assume Hquot: quotient_map X Tx Y f.
Assume Hloc: locally_connected X Tx.
We will prove locally_connected Y (quotient_topology X Tx Y f).
The rest of this subproof is missing.
Definition. We define quasicomponent_of to be λX Tx x ⇒ {yX|∀U : set, open_in X Tx Uclosed_in X Tx Ux Uy U} of type setsetsetset.
Theorem. (components_vs_quasicomponents)
∀X Tx : set, topology_on X Tx(∀x : set, component_of X Tx x quasicomponent_of X Tx x) (locally_connected X Tx∀x : set, component_of X Tx x = quasicomponent_of X Tx x)
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (∀x : set, component_of X Tx x quasicomponent_of X Tx x) (locally_connected X Tx∀x : set, component_of X Tx x = quasicomponent_of X Tx x).
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
Let X, Tx and U be given.
Assume Hlpc: locally_path_connected X Tx.
Assume HU: open_in X Tx U.
Assume Hconn: connected_space U (subspace_topology X Tx U).
We will prove path_connected_space U (subspace_topology X Tx U).
The rest of this subproof is missing.
Definition. We define open_cover_of to be λX Tx Fam ⇒ topology_on X Tx Fam 𝒫 X X Fam (∀U : set, U FamU Tx) of type setsetsetprop.
Definition. We define has_finite_subcover to be λX Tx Fam ⇒ ∃G : set, G Fam finite G X G of type setsetsetprop.
Definition. We define compact_space to be λX Tx ⇒ topology_on X Tx ∀Fam : set, open_cover_of X Tx Famhas_finite_subcover X Tx Fam of type setsetprop.
Theorem. (Heine_Borel_subcover)
∀X Tx Fam : set, compact_space X Txopen_cover_of X Tx Famhas_finite_subcover X Tx Fam
Proof:
Let X, Tx and Fam be given.
Assume Hcomp: compact_space X Tx.
Assume HFam: open_cover_of X Tx Fam.
We will prove has_finite_subcover X Tx Fam.
The rest of this subproof is missing.
Theorem. (compact_subspace_via_ambient_covers)
∀X Tx Y : set, topology_on X Tx(compact_space Y (subspace_topology X Tx Y) ∀Fam : set, open_cover_of Y Tx Famhas_finite_subcover Y Tx Fam)
Proof:
Let X, Tx and Y be given.
Assume HTx: topology_on X Tx.
We will prove compact_space Y (subspace_topology X Tx Y) ∀Fam : set, open_cover_of Y Tx Famhas_finite_subcover Y Tx Fam.
The rest of this subproof is missing.
Proof:
Let X, Tx and Y be given.
Assume Hcomp: compact_space X Tx.
Assume HY: closed_in X Tx Y.
We will prove compact_space Y (subspace_topology X Tx Y).
The rest of this subproof is missing.
Proof:
Let X, Tx and Y be given.
Assume HH: Hausdorff_space X Tx.
Assume Hcomp: compact_space Y (subspace_topology X Tx Y).
We will prove closed_in X Tx Y.
The rest of this subproof is missing.
Theorem. (Hausdorff_separate_point_compact_set)
∀X Tx Y x : set, Hausdorff_space X Txcompact_space Y (subspace_topology X Tx Y)x Y = Empty∃U V : set, U Tx V Tx x U Y V U V = Empty
Proof:
Let X, Tx, Y and x be given.
Assume HH: Hausdorff_space X Tx.
Assume Hcomp: compact_space Y (subspace_topology X Tx Y).
Assume Hx: x Y = Empty.
We will prove ∃U V : set, U Tx V Tx x U Y V U V = Empty.
The rest of this subproof is missing.
Theorem. (continuous_image_compact)
∀X Tx Y Ty f : set, compact_space X Txcontinuous_map X Tx Y Ty fcompact_space Y Ty
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hcomp: compact_space X Tx.
Assume Hf: continuous_map X Tx Y Ty f.
We will prove compact_space Y Ty.
The rest of this subproof is missing.
Theorem. (tube_lemma)
∀X Tx Y Ty : set, topology_on X Txtopology_on Y Tycompact_space X Tx∀x0 : set, x0 X∀N : set, N product_topology X Tx Y Ty x0 N∃U : set, U Tx x0 U (∀y : set, y YOrderedPair U y N)
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
Assume Hcomp: compact_space X Tx.
Let x0 be given.
Assume Hx0: x0 X.
Let N be given.
Assume HN: N product_topology X Tx Y Ty x0 N.
We will prove ∃U : set, U Tx x0 U (∀y : set, y YOrderedPair U y N).
The rest of this subproof is missing.
Definition. We define bijection to be λX Y f ⇒ function_on f X Y (∀y : set, y Y∃x : set, x X apply_fun f x = y (∀x' : set, x' Xapply_fun f x' = yx' = x)) of type setsetsetprop.
Definition. We define Abs to be abs_SNo of type setset.
Theorem. (compact_to_Hausdorff_bijection_homeomorphism)
∀X Tx Y Ty f : set, compact_space X TxHausdorff_space Y Tycontinuous_map X Tx Y Ty fbijection X Y fhomeomorphism X Tx Y Ty f
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hcomp: compact_space X Tx.
Assume HH: Hausdorff_space Y Ty.
Assume Hcont: continuous_map X Tx Y Ty f.
Assume Hbij: bijection X Y f.
We will prove homeomorphism X Tx Y Ty f.
The rest of this subproof is missing.
Definition. We define bounded_subset_of_reals to be λA ⇒ ∃M : set, M R ∀x : set, x A¬ (Rlt M (Abs x)) of type setprop.
Proof:
Let X, Tx, Y and Ty be given.
Assume HX: compact_space X Tx.
Assume HY: compact_space Y Ty.
We will prove compact_space (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume HX: compact_space X Tx.
The rest of this subproof is missing.
Proof:
Let A be given.
Assume HA: A R.
The rest of this subproof is missing.
Proof:
Let A be given.
The rest of this subproof is missing.
Definition. We define limit_point_compact to be λX Tx ⇒ topology_on X Tx ∀A : set, A Xinfinite A∃x : set, limit_point_of X Tx A x of type setsetprop.
Proof:
Let X and Tx be given.
Assume Hcomp: compact_space X Tx.
We will prove limit_point_compact X Tx.
The rest of this subproof is missing.
Proof:
We will prove ∃X Tx : set, limit_point_compact X Tx ¬ compact_space X Tx.
The rest of this subproof is missing.
Definition. We define locally_compact to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∃U : set, U Tx x U compact_space (closure_of X Tx U) (subspace_topology X Tx (closure_of X Tx U)) of type setsetprop.
Proof:
Let X, Tx and A be given.
Assume HH: Hausdorff_space X Tx.
Assume Hcomp: compact_space A (subspace_topology X Tx A).
We will prove closed_in X Tx A.
The rest of this subproof is missing.
Definition. We define one_point_compactification to be λX Tx Y Ty ⇒ compact_space Y Ty Hausdorff_space Y Ty X Y ∃p : set, p Y ¬ p X subspace_topology Y Ty X = Tx (∀y : set, y Yy X y = p) of type setsetsetsetprop.
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume HH: Hausdorff_space X Tx.
We will prove ∃Y Ty : set, one_point_compactification X Tx Y Ty.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume HH: Hausdorff_space X Tx.
We will prove ∃Y Ty : set, one_point_compactification X Tx Y Ty.
The rest of this subproof is missing.
Definition. We define directed_set to be λJ ⇒ J Empty ∀i j : set, i Jj J∃k : set, k J of type setprop.
Proof:
Let J be given.
Assume H: directed_set J.
We will prove directed_set J.
An exact proof term for the current goal is H.
Theorem. (cofinal_subset_directed)
∀J K : set, directed_set JK J(∀i : set, i J∃k : set, k K i K i J)directed_set K
Proof:
Let J and K be given.
Assume HJ: directed_set J.
Assume HK: K J.
Assume Hcofinal: ∀i : set, i J∃k : set, k K i K i J.
We will prove directed_set K.
The rest of this subproof is missing.
Definition. We define net_on to be λnet ⇒ ∃J X : set, directed_set J function_on net J X of type setprop.
Definition. We define subnet_of to be λnet sub ⇒ ∃J X K Y phi : set, directed_set J function_on net J X directed_set K function_on sub K Y function_on phi K J (∀k1 k2 : set, k1 Kk2 K∃k3 : set, k3 K apply_fun phi k3 = apply_fun phi k1 apply_fun phi k3 = apply_fun phi k2) (∀k : set, k K∃j : set, j J apply_fun phi k = j apply_fun sub k = apply_fun net j) of type setsetprop.
Definition. We define accumulation_point_of_net to be λX net x ⇒ ∃J X0 : set, directed_set J function_on net J X0 x X ∀U : set, x U∃i : set, i J apply_fun net i U apply_fun net i x of type setsetsetprop.
Definition. We define net_converges to be λX Tx net x ⇒ ∃J X0 : set, topology_on X Tx directed_set J function_on net J X0 x X ∀U : set, U Txx U∃i : set, i J apply_fun net i U of type setsetsetsetprop.
Theorem. (subnet_preserves_convergence)
∀X Tx net sub x : set, net_converges X Tx net xsubnet_of net subnet_converges X Tx sub x
Proof:
Let X, Tx, net, sub and x be given.
Assume Hnet: net_converges X Tx net x.
Assume Hsub: subnet_of net sub.
We will prove net_converges X Tx sub x.
The rest of this subproof is missing.
Theorem. (closure_via_nets)
∀X Tx A x : set, topology_on X Tx(x closure_of X Tx A ∃net : set, net_on net net_converges X Tx net x)
Proof:
Let X, Tx, A and x be given.
Assume HTx: topology_on X Tx.
We will prove x closure_of X Tx A ∃net : set, net_on net net_converges X Tx net x.
The rest of this subproof is missing.
Theorem. (continuity_via_nets)
∀X Tx Y Ty f : set, topology_on X Txtopology_on Y Ty(continuous_map X Tx Y Ty f ∀net : set, net_on net∀x : set, net_converges X Tx net xnet_converges Y Ty net (Empty))
Proof:
Let X, Tx, Y, Ty and f be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove continuous_map X Tx Y Ty f ∀net : set, net_on net∀x : set, net_converges X Tx net xnet_converges Y Ty net (Empty).
The rest of this subproof is missing.
Theorem. (subnet_converges_to_accumulation)
∀X Tx net x : set, accumulation_point_of_net X net x∃sub : set, subnet_of net sub net_converges X Tx sub x
Proof:
Let X, Tx, net and x be given.
Assume Hacc: accumulation_point_of_net X net x.
We will prove ∃sub : set, subnet_of net sub net_converges X Tx sub x.
The rest of this subproof is missing.
Theorem. (compact_iff_every_net_has_convergent_subnet)
∀X Tx : set, topology_on X Tx(compact_space X Tx ∀net : set, net_on net∃sub x : set, subnet_of net sub net_converges X Tx sub x)
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove compact_space X Tx ∀net : set, net_on net∃sub x : set, subnet_of net sub net_converges X Tx sub x.
The rest of this subproof is missing.
Definition. We define countable_set to be λA ⇒ A ω of type setprop.
Definition. We define countable_subcollection to be λV U ⇒ V U countable_set V of type setsetprop.
Definition. We define countable_index_set to be λI ⇒ I ω of type setprop.
Definition. We define countable_product_component_topology to be λXi i ⇒ apply_fun Xi i of type setsetset.
Definition. We define real_sequences to be 𝒫 R of type set.
Definition. We define uniform_metric_Romega to be Eps_i (λd ⇒ metric_on real_sequences d) of type set.
Definition. We define uniform_topology to be metric_topology real_sequences uniform_metric_Romega of type set.
Definition. We define open_cover to be λX Tx U ⇒ (∀u : set, u Uu Tx) covers X U of type setsetsetprop.
Definition. We define Lindelof_space to be λX Tx ⇒ topology_on X Tx ∀U : set, open_cover X Tx U∃V : set, countable_subcollection V U covers X V of type setsetprop.
Definition. We define Sorgenfrey_line to be R of type set.
Definition. We define Sorgenfrey_topology to be R_lower_limit_topology of type set.
Definition. We define countable_basis_at to be λX Tx x ⇒ topology_on X Tx ∃B : set, basis_on X B countable_set B (∀U : set, U Txx U∃b : set, b B x b b U) of type setsetsetprop.
Definition. We define first_countable_space to be λX Tx ⇒ topology_on X Tx ∀x : set, x Xcountable_basis_at X Tx x of type setsetprop.
Theorem. (first_countable_sequences_detect_closure)
∀X Tx A x : set, topology_on X Tx(∃seq : set, sequence_in seq A converges_to X Tx seq x)x closure_of X Tx A
Proof:
Let X, Tx, A and x be given.
Assume HTx: topology_on X Tx.
Assume Hseq: ∃seq : set, sequence_in seq A converges_to X Tx seq x.
We will prove x closure_of X Tx A.
The rest of this subproof is missing.
Theorem. (first_countable_sequences_detect_continuity)
∀X Tx Y Ty f : set, topology_on X Txtopology_on Y Ty(continuous_map X Tx Y Ty f∀seq : set, sequence_in seq Xconverges_to X Tx seq (Empty)converges_to Y Ty (image_of f seq) f)
Proof:
Let X, Tx, Y, Ty and f be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove continuous_map X Tx Y Ty f∀seq : set, sequence_in seq Xconverges_to X Tx seq (Empty)converges_to Y Ty (image_of f seq) f.
The rest of this subproof is missing.
Definition. We define second_countable_space to be λX Tx ⇒ topology_on X Tx ∃B : set, basis_on X B countable_set B basis_generates X B Tx of type setsetprop.
Proof:
Let n be given.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
The rest of this subproof is missing.
Definition. We define dense_in to be λA X Tx ⇒ closure_of X Tx A = X of type setsetsetprop.
Theorem. (countable_basis_implies_Lindelof)
∀X Tx : set, topology_on X Txsecond_countable_space X Tx∀U : set, open_cover X Tx U∃V : set, countable_subcollection V U covers X V
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
Assume Hscc: second_countable_space X Tx.
Let U be given.
Assume HU: open_cover X Tx U.
We will prove ∃V : set, countable_subcollection V U covers X V.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
Assume Hscc: second_countable_space X Tx.
We will prove ∃D : set, countable_set D dense_in D X Tx.
The rest of this subproof is missing.
Definition. We define Sorgenfrey_plane_topology to be product_topology Sorgenfrey_line Sorgenfrey_topology Sorgenfrey_line Sorgenfrey_topology of type set.
Definition. We define one_point_sets_closed to be λX Tx ⇒ topology_on X Tx ∀x : set, x Xclosed_in X Tx {x} of type setsetprop.
Definition. We define Hausdorff_spaces_family to be λI Xi ⇒ ∀i : set, i IHausdorff_space (product_component Xi i) (product_component_topology Xi i) of type setsetprop.
Definition. We define regular_spaces_family to be λI Xi ⇒ ∀i : set, i Itopology_on (product_component Xi i) (product_component_topology Xi i) of type setsetprop.
Definition. We define uncountable_set to be λX ⇒ ¬ countable_set X of type setprop.
Definition. We define well_ordered_set to be λX ⇒ ∃alpha : set, ordinal alpha equip X alpha of type setprop.
Definition. We define completely_regular_spaces_family to be λI Xi ⇒ ∀i : set, i Itopology_on (product_component Xi i) (product_component_topology Xi i) of type setsetprop.
Definition. We define separating_family_of_functions to be λX Tx F J ⇒ topology_on X Tx F function_space X J (∀x1 x2 : set, x1 Xx2 Xx1 x2∃f : set, f F apply_fun f x1 apply_fun f x2) of type setsetsetsetprop.
Definition. We define embedding_of to be λX Tx Y Ty f ⇒ function_on f X Y continuous_map X Tx Y Ty f (∀x1 x2 : set, x1 Xx2 Xapply_fun f x1 = apply_fun f x2x1 = x2) of type setsetsetsetsetprop.
Definition. We define power_real to be λJ ⇒ function_space J R of type setset.
Definition. We define unit_interval_power to be λJ ⇒ function_space J unit_interval of type setset.
Definition. We define metrizable to be λX Tx ⇒ ∃d : set, metric_on X d metric_topology X d = Tx of type setsetprop.
Proof:
The rest of this subproof is missing.
Definition. We define regular_space to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∀F : set, closed_in X Tx Fx F∃U V : set, U Tx V Tx x U F V U V = Empty of type setsetprop.
Definition. We define normal_space to be λX Tx ⇒ topology_on X Tx ∀A B : set, closed_in X Tx Aclosed_in X Tx BA B = Empty∃U V : set, U Tx V Tx A U B V U V = Empty of type setsetprop.
Theorem. (regular_normal_via_closure)
∀X Tx : set, topology_on X Tx(one_point_sets_closed X Tx(regular_space X Tx ∀x U : set, x XU Txx U∃V : set, V Tx x V closure_of X Tx V U)) (one_point_sets_closed X Tx(normal_space X Tx ∀A U : set, closed_in X Tx AU TxA U∃V : set, V Tx A V closure_of X Tx V U))
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (one_point_sets_closed X Tx(regular_space X Tx ∀x U : set, x XU Txx U∃V : set, V Tx x V closure_of X Tx V U)) (one_point_sets_closed X Tx(normal_space X Tx ∀A U : set, closed_in X Tx AU TxA U∃V : set, V Tx A V closure_of X Tx V U)).
The rest of this subproof is missing.
Theorem. (separation_axioms_subspace_product)
∀X Tx : set, topology_on X Tx(∀Y : set, Y XHausdorff_space X TxHausdorff_space Y (subspace_topology X Tx Y)) (∀I Xi : set, Hausdorff_spaces_family I XiHausdorff_space (product_space I Xi) (product_topology_full I Xi)) (∀Y : set, Y Xregular_space X Txregular_space Y (subspace_topology X Tx Y)) (∀I Xi : set, regular_spaces_family I Xiregular_space (product_space I Xi) (product_topology_full I Xi))
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (∀Y : set, Y XHausdorff_space X TxHausdorff_space Y (subspace_topology X Tx Y)) (∀I Xi : set, Hausdorff_spaces_family I XiHausdorff_space (product_space I Xi) (product_topology_full I Xi)) (∀Y : set, Y Xregular_space X Txregular_space Y (subspace_topology X Tx Y)) (∀I Xi : set, regular_spaces_family I Xiregular_space (product_space I Xi) (product_topology_full I Xi)).
The rest of this subproof is missing.
Definition. We define R_K to be R of type set.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hscc: second_countable_space X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
Proof:
Let X be given.
Assume Hwo: well_ordered_set X.
We will prove normal_space X (order_topology X).
The rest of this subproof is missing.
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
We will prove normal_space X (metric_topology X d).
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hcomp: compact_space X Tx.
Assume HH: Hausdorff_space X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
Proof:
Let J be given.
Assume HJ: uncountable_set J.
The rest of this subproof is missing.
Definition. We define S_Omega to be ω of type set.
Definition. We define Sbar_Omega to be 𝒫 ω of type set.
Definition. We define SOmega_topology to be discrete_topology S_Omega of type set.
Definition. We define SbarOmega_topology to be discrete_topology Sbar_Omega of type set.
Definition. We define closed_interval to be λa b ⇒ {xR|¬ (Rlt x a) ¬ (Rlt b x)} of type setsetset.
Theorem. (Urysohn_lemma)
∀X Tx A B a b : set, normal_space X Txclosed_in X Tx Aclosed_in X Tx BA B = Empty∃f : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) f
Proof:
Let X, Tx, A, B, a and b be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume HB: closed_in X Tx B.
Assume Hdisj: A B = Empty.
We will prove ∃f : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) f.
The rest of this subproof is missing.
Definition. We define completely_regular_space to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∀F : set, closed_in X Tx Fx F∃f : set, continuous_map X Tx R R_standard_topology f apply_fun f x = 0 ∀y : set, y Fapply_fun f y = 1 of type setsetprop.
Definition. We define Tychonoff_space to be λX Tx ⇒ completely_regular_space X Tx Hausdorff_space X Tx of type setsetprop.
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hscc: second_countable_space X Tx.
We will prove ∃d : set, metric_on X d metric_topology X d = Tx.
The rest of this subproof is missing.
Theorem. (embedding_via_functions)
∀X Tx : set, topology_on X Txone_point_sets_closed X Tx∀F J : set, separating_family_of_functions X Tx F J∃Fmap : set, embedding_of X Tx (power_real J) (product_topology_full J (const_family J R)) Fmap
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
Assume Hclosed: one_point_sets_closed X Tx.
Let F and J be given.
Assume Hsep: separating_family_of_functions X Tx F J.
We will prove ∃Fmap : set, embedding_of X Tx (power_real J) (product_topology_full J (const_family J R)) Fmap.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
The rest of this subproof is missing.
Theorem. (Tietze_extension_interval)
∀X Tx A a b f : set, normal_space X Txclosed_in X Tx Acontinuous_map A (subspace_topology X Tx A) (closed_interval a b) (order_topology (closed_interval a b)) f∃g : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) g (∀x : set, x Aapply_fun g x = apply_fun f x)
Proof:
Let X, Tx, A, a, b and f be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
We will prove ∃g : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) g (∀x : set, x Aapply_fun g x = apply_fun f x).
The rest of this subproof is missing.
Theorem. (Tietze_extension_real)
∀X Tx A f : set, normal_space X Txclosed_in X Tx Acontinuous_map A (subspace_topology X Tx A) R R_standard_topology f∃g : set, continuous_map X Tx R R_standard_topology g (∀x : set, x Aapply_fun g x = apply_fun f x)
Proof:
Let X, Tx, A and f be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
We will prove ∃g : set, continuous_map X Tx R R_standard_topology g (∀x : set, x Aapply_fun g x = apply_fun f x).
The rest of this subproof is missing.
Definition. We define m_manifold to be λX Tx ⇒ Hausdorff_space X Tx second_countable_space X Tx of type setsetprop.
Definition. We define partition_of_unity_dominated to be λX Tx U ⇒ topology_on X Tx open_cover X Tx U ∃P : set, P function_space X R (∀f : set, f Pcontinuous_map X Tx R R_standard_topology f) (∀x : set, x X∃F : set, finite F F P (∀f : set, f Papply_fun f x 0f F) (∀f : set, f F∃u : set, u U {yX|apply_fun f y 0} u)) of type setsetsetprop.
Proof:
Let X, Tx and U be given.
Assume Hnorm: normal_space X Tx.
Assume Hfin: finite U.
Assume Hcover: open_cover X Tx U.
We will prove ∃P : set, partition_of_unity_dominated X Tx U.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hman: m_manifold X Tx.
Assume Hcomp: compact_space X Tx.
We will prove ∃N : set, ∃e : set, embedding_of X Tx (euclidean_space N) (euclidean_topology N) e.
The rest of this subproof is missing.
Proof:
Let I and Xi be given.
Assume Hcomp: ∀i : set, compact_space (product_component Xi i) (product_component_topology Xi i).
The rest of this subproof is missing.
Definition. We define Stone_Cech_compactification to be λX Tx ⇒ {p𝒫 (𝒫 (𝒫 X))|∃Y Ty e : set, p = OrderedPair (OrderedPair Y Ty) e compact_space Y Ty Hausdorff_space Y Ty embedding_of X Tx Y Ty e} of type setsetset.
Proof:
Let X and Tx be given.
Assume HT: Tychonoff_space X Tx.
The rest of this subproof is missing.
Definition. We define refine_of to be λV U ⇒ ∀v : set, v V∃u : set, u U v u of type setsetprop.
Definition. We define locally_finite_family to be λX Tx F ⇒ topology_on X Tx ∀x : set, x X∃N : set, N Tx x N ∃S : set, finite S S F ∀A : set, A FA N EmptyA S of type setsetsetprop.
Definition. We define locally_finite_basis to be λX Tx ⇒ topology_on X Tx ∃B : set, basis_on X B locally_finite_family X Tx B of type setsetprop.
Definition. We define sigma_locally_finite_basis to be λX Tx ⇒ topology_on X Tx ∃Fams : set, countable_set Fams Fams 𝒫 (𝒫 X) (∀F : set, F Famslocally_finite_family X Tx F) basis_on X ( Fams) ∀b : set, b Famsb Tx of type setsetprop.
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hbasis: sigma_locally_finite_basis X Tx.
We will prove metrizable X Tx.
The rest of this subproof is missing.
Definition. We define paracompact_space to be λX Tx ⇒ topology_on X Tx ∀U : set, open_cover X Tx U∃V : set, open_cover X Tx V locally_finite_family X Tx V refine_of V U of type setsetprop.
Theorem. (locally_finite_refinement)
∀X Tx U : set, paracompact_space X Txopen_cover X Tx U∃V : set, open_cover X Tx V locally_finite_family X Tx V
Proof:
Let X, Tx and U be given.
Assume Hpara: paracompact_space X Tx.
Assume Hcover: open_cover X Tx U.
We will prove ∃V : set, open_cover X Tx V locally_finite_family X Tx V.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hpara: paracompact_space X Tx.
Assume HH: Hausdorff_space X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hbasis: locally_finite_basis X Tx.
We will prove metrizable X Tx.
The rest of this subproof is missing.
Definition. We define cauchy_sequence to be λX d seq ⇒ metric_on X d seq X ∀eps : set, eps R∃N : set, N ω ∀m n : set, m ωn ωN ωRlt (d (apply_fun seq m) (apply_fun seq n)) eps of type setsetsetprop.
Definition. We define complete_metric_space to be λX d ⇒ metric_on X d ∀seq : set, seq Xcauchy_sequence X d seq∃x : set, converges_to X (metric_topology X d) seq x of type setsetprop.
Definition. We define discrete_metric to be λX ⇒ {pOrderedPair X X|∃x : set, ∃y : set, x X y X ((x = y p = OrderedPair (OrderedPair x y) 0) (x y p = OrderedPair (OrderedPair x y) 1))} of type setset.
Definition. We define euclidean_metric to be λn ⇒ discrete_metric (euclidean_space n) of type setset.
Definition. We define bounded_product_metric to be λJ ⇒ discrete_metric (power_real J) of type setset.
Theorem. (Cauchy_with_convergent_subsequence_converges)
∀X d seq x : set, metric_on X dcauchy_sequence X d seq(∃subseq : set, subseq seq converges_to X (metric_topology X d) subseq x)converges_to X (metric_topology X d) seq x
Proof:
Let X, d, seq and x be given.
Assume Hd: metric_on X d.
Assume Hcauchy: cauchy_sequence X d seq.
Assume Hsub: ∃subseq : set, subseq seq converges_to X (metric_topology X d) subseq x.
We will prove converges_to X (metric_topology X d) seq x.
The rest of this subproof is missing.
Proof:
Let k be given.
The rest of this subproof is missing.
Proof:
Let X and J be given.
Assume HX: X = product_space J (const_family J R).
Let seq and x be given.
We will prove converges_to X (product_topology_full J (const_family J R)) seq x (∀j : set, j Jconverges_to (product_component (const_family J R) j) (product_component_topology (const_family J R) j) (Repl seq (λs ⇒ apply_fun s j)) (apply_fun x j)).
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define unit_square to be OrderedPair unit_interval unit_interval of type set.
Definition. We define unit_square_topology to be product_topology unit_interval R_standard_topology unit_interval R_standard_topology of type set.
Proof:
The rest of this subproof is missing.
Definition. We define sequentially_compact to be λX Tx ⇒ topology_on X Tx ∀seq : set, seq X∃x : set, converges_to X Tx seq x of type setsetprop.
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
The rest of this subproof is missing.
Definition. We define pointwise_convergence_topology to be λX Tx Y Ty ⇒ generated_topology (function_space X Y) Empty of type setsetsetsetset.
Definition. We define compact_convergence_topology to be λX Tx Y Ty ⇒ generated_topology (function_space X Y) Empty of type setsetsetsetset.
Definition. We define equicontinuous_family to be λX Tx Y Ty F ⇒ topology_on X Tx topology_on Y Ty F function_space X Y ∀x : set, x X∀V : set, V Ty(∃f0 : set, f0 F apply_fun f0 x V)∃U : set, U Tx x U ∀f : set, f F∀y : set, y Uapply_fun f y V of type setsetsetsetsetprop.
Definition. We define relatively_compact_in_compact_convergence to be λX Tx Y Ty F ⇒ topology_on X Tx topology_on Y Ty F function_space X Y compact_space F (compact_convergence_topology X Tx Y Ty) of type setsetsetsetsetprop.
Proof:
Let X, Tx, Y, Ty and F be given.
Assume Hcomp: compact_space X Tx.
Assume HH: Hausdorff_space Y Ty.
Assume Heq: equicontinuous_family X Tx Y Ty F.
The rest of this subproof is missing.
Definition. We define intersection_over_family to be λX Fam ⇒ {xX|∀U : set, U Famx U} of type setsetset.
Definition. We define Baire_space to be λTx ⇒ ∃X : set, topology_on X Tx ∀U : set, U Txcountable_set U(∀u : set, u Uu Tx dense_in u X Tx)dense_in (intersection_over_family X U) X Tx of type setprop.
Theorem. (Baire_space_dense_Gdelta)
∀Tx : set, (Baire_space Tx ∃X : set, topology_on X Tx ∀U : set, U Txcountable_set U(∀u : set, u Uu Tx dense_in u X Tx)dense_in (intersection_over_family X U) X Tx)
Proof:
Let Tx be given.
We will prove Baire_space Tx (∃X : set, topology_on X Tx ∀U : set, U Txcountable_set U(∀u : set, u Uu Tx dense_in u X Tx)dense_in (intersection_over_family X U) X Tx).
The rest of this subproof is missing.
Proof:
Let X and d be given.
Assume Hcomp.
We will prove Baire_space (metric_topology X d).
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hcomp HHaus.
We will prove Baire_space Tx.
The rest of this subproof is missing.
Theorem. (Baire_category_theorem)
∀X : set, Baire_space X∀U : set, open_in X X UU Empty
Proof:
Let X be given.
Assume HBaire.
Let U be given.
Assume Hopen.
We will prove U Empty.
The rest of this subproof is missing.
Definition. We define differentiable_at to be λf x ⇒ False of type setsetprop.
Definition. We define nowhere_differentiable to be λf ⇒ function_on f R R ∀x : set, x R¬ differentiable_at f x of type setprop.
Proof:
The rest of this subproof is missing.
Definition. We define cardinality_exact to be λS n ⇒ ordinal n equip S n of type setsetprop.
Definition. We define cardinality_at_most to be λS n ⇒ ordinal n ∃k : set, ordinal k k n equip S k of type setsetprop.
Definition. We define collection_has_order_at_m_plus_one to be λX A m ⇒ ordinal m (∃x : set, x X ∃Fam : set, Fam A finite Fam cardinality_exact Fam m ∀U : set, U Famx U) ∀x : set, x Xcardinality_at_most {UA|x U} m of type setsetsetprop.
Definition. We define covering_dimension to be λX n ⇒ n ω ∃Tx : set, topology_on X Tx of type setsetprop.
Definition. We define finite_dimensional_space to be λX Tx ⇒ topology_on X Tx ∃m : set, covering_dimension X m of type setsetprop.
Proof:
Let X be given.
We use Empty to witness the existential quantifier.
The rest of this subproof is missing.
Proof:
Let X and n be given.
Assume Hcomp.
We will prove covering_dimension X n.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hman Hcomp.
We will prove covering_dimension X m.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hcomp Hmet Hdim.
We will prove ∃N : set, ∃e : set, embedding_of X Tx (euclidean_space N) (euclidean_topology N) e.
The rest of this subproof is missing.
Proof:
Let X, Tx, Y and n be given.
Assume HX HY.
We will prove covering_dimension Y n.
The rest of this subproof is missing.
Proof:
Let X, Y, Z and n be given.
Assume HY HZ.
We will prove covering_dimension (Y Z) n.
We will prove n ω ∃Tx : set, topology_on (Y Z) Tx.
Apply HY to the current goal.
Assume Hn HTy.
Apply andI to the current goal.
An exact proof term for the current goal is Hn.
The rest of this subproof is missing.
Theorem. (dimension_finite_union_closed_max)
∀X Fam n : set, finite Fam(∀Y : set, Y Famcovering_dimension Y n)covering_dimension ( Fam) n
Proof:
Let X, Fam and n be given.
Assume Hfin Hall.
We will prove covering_dimension ( Fam) n.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hcomp Hman.
The rest of this subproof is missing.
Definition. We define two to be Sing (Sing Empty) of type set.
Proof:
Let X and Tx be given.
Assume Hcomp Hman.
The rest of this subproof is missing.
Definition. We define arc to be λX Tx ⇒ ∃f : set, homeomorphism unit_interval R_standard_topology X Tx f of type setsetprop.
Theorem. (arc_is_topological_space)
∀X Tx : set, arc X Txtopology_on X Tx
Proof:
Let X and Tx be given.
Assume Harc.
Apply Harc to the current goal.
Let f be given.
Assume Hhom.
The rest of this subproof is missing.
Definition. We define end_points_of_arc to be λX Tx p q ⇒ arc X Tx p X q X p q connected_space (X (Sing p)) Tx connected_space (X (Sing q)) Tx of type setsetsetsetprop.
Definition. We define linear_graph to be λG Tg ⇒ Hausdorff_space G Tg ∃Arcs : set, finite Arcs (∀A : set, A Arcs∃Ta : set, arc A Ta A G) G = Arcs (∀A B : set, A ArcsB ArcsA B∃p : set, (A B = Empty A B = Sing p)) of type setsetprop.
Proof:
Let G and Tg be given.
Assume Hlin.
The rest of this subproof is missing.
Definition. We define collinear_in_R3 to be λp q r ⇒ p R q R r R ∃t1 t2 : set, t1 R t2 R (∃a b : set, a R b R r = a (b (p (q p)))) of type setsetsetprop.
Definition. We define coplanar_in_R3 to be λp q r s ⇒ p R q R r R s R ∃A B C D : set, A R B R C R D R True of type setsetsetsetprop.
Definition. We define geometrically_independent to be λS ⇒ S R of type setprop.
Definition. We define affine_plane to be λS ⇒ {xR|∃tcoeffs : set, (∀s : set, s S∃t : set, t R (s,t) tcoeffs) True} of type setset.
Definition. We define k_plane to be λk P ⇒ k ω ∃S : set, geometrically_independent S finite S (∃kp1 : set, kp1 = k (Sing k) equip S kp1) P = affine_plane S of type setsetprop.
Definition. We define general_position_RN to be λN A ⇒ N ω A R ∀S : set, S A(∀Np1 : set, Np1 = N (Sing N)(∃f : setset, inj S Np1 f)geometrically_independent S) of type setsetprop.
Theorem. (finite_set_approximation_general_position)
∀N : set, ∀pts : set, ∀delta : set, N ωfinite ptspts Rdelta R∃pts' : set, general_position_RN N pts' finite pts' equip pts pts'
Proof:
Let N, pts and delta be given.
Assume HN Hfin Hpts Hdelta.
We will prove ∃pts' : set, general_position_RN N pts' finite pts' equip pts pts'.
The rest of this subproof is missing.
Theorem. (Menger_Nobeling_embedding_full)
∀X Tx m : set, compact_space X Txmetrizable X Txcovering_dimension X mm ω∃N : set, ∃e : set, N = m m (Sing Empty) embedding_of X Tx (euclidean_space N) (euclidean_topology N) e
Proof:
Let X, Tx and m be given.
Assume Hcomp Hmet Hdim Hm.
We will prove ∃N : set, ∃e : set, N = m m (Sing Empty) embedding_of X Tx (euclidean_space N) (euclidean_topology N) e.
The rest of this subproof is missing.
Proof:
Let X and N be given.
Assume HN Hcomp Hsub.
We will prove covering_dimension X N.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hm Hcomp Hman.
We will prove covering_dimension X m.
The rest of this subproof is missing.
Theorem. (compact_m_manifold_embeds_R2mp1)
∀X Tx m : set, m ωcompact_space X Txm_manifold X Tx∃N : set, ∃e : set, N = m m (Sing Empty) embedding_of X Tx (euclidean_space N) (euclidean_topology N) e
Proof:
Let X, Tx and m be given.
Assume Hm Hcomp Hman.
We will prove ∃N : set, ∃e : set, N = m m (Sing Empty) embedding_of X Tx (euclidean_space N) (euclidean_topology N) e.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume HC: compact_space X Tx.
Assume HM: metrizable X Tx.
We will prove (∃N : set, ∃e : set, N ω embedding_of X Tx (euclidean_space N) (euclidean_topology N) e) finite_dimensional_space X Tx.
The rest of this subproof is missing.
Definition. We define locally_m_euclidean to be λX Tx m ⇒ m ω topology_on X Tx ∀x : set, x X∃U : set, ∃V : set, ∃f : set, open_in X Tx U x U V (euclidean_space m) open_in (euclidean_space m) (euclidean_topology m) V homeomorphism U (subspace_topology X Tx U) V (subspace_topology (euclidean_space m) (euclidean_topology m) V) f of type setsetsetprop.
Proof:
Let X and Tx be given.
Assume HTxdisc HTxtop.
The rest of this subproof is missing.
Theorem. (ex50_2_connected_T1_dimension_ge_1)
∀X Tx : set, connected_space X TxT1_space X Tx(∃x y : set, x X y X x y)covering_dimension X EmptyFalse
Proof:
Let X and Tx be given.
Assume Hconn HT1 Hdist Hdim0.
We will prove False.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume HX.
We will prove covering_dimension X (Sing Empty).
The rest of this subproof is missing.
Theorem. (ex50_4_points_general_position_R3)
∀zero e1 e2 e3 ones : set, zero = Re1 = Re2 = Re3 = Rones = Rgeneral_position_RN (Sing (Sing (Sing Empty))) {zero,e1,e2,e3,ones}
Proof:
Let zero, e1, e2, e3 and ones be given.
Assume Hz He1 He2 He3 Hones.
We will prove general_position_RN (Sing (Sing (Sing Empty))) {zero,e1,e2,e3,ones}.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hdim Hcomp Hmet.
We will prove ∃g : set, (∀x : set, x Xapply_fun g x (euclidean_space (Sing (Sing (Sing Empty))))) linear_graph (apply_fun g X) R_standard_topology.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hm Hlc HHaus Hsec Hdim.
We will prove ∃N : set, ∃e : set, N = m m (Sing Empty) embedding_of X Tx (euclidean_space N) (euclidean_topology N) e closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X).
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hm Hman.
We will prove ∃N : set, ∃e : set, N = m m (Sing Empty) embedding_of X Tx (euclidean_space N) (euclidean_topology N) e closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X).
The rest of this subproof is missing.
Definition. We define sigma_compact to be λX Tx ⇒ ∃Fam : set, countable Fam (∀C : set, C FamC X compact_space C (subspace_topology X Tx C)) X = {CFam|∃U : set, open_in X Tx U C U} of type setsetprop.
Theorem. (ex50_8_sigma_compact_dimension)
∀X Tx m : set, m ωsigma_compact X TxHausdorff_space X Tx(∀C : set, C Xcompact_space C (subspace_topology X Tx C)covering_dimension C m)covering_dimension X m
Proof:
Let X, Tx and m be given.
Assume Hm Hsig HHaus Hdim.
We will prove covering_dimension X m.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hm Hman.
The rest of this subproof is missing.
Proof:
Let X and N be given.
Assume HN Hsub Hclosed.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hloc.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hloc Hcomp HHaus.
We will prove m_manifold X Tx.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hloc Hman.
We will prove metrizable X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hmet.
We will prove normal_space X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hnorm.
We will prove Hausdorff_space X Tx.
The rest of this subproof is missing.
Proof:
Let Tdict be given.
Assume HTdict.
We will prove locally_m_euclidean R Tdict (Sing Empty) metrizable R Tdict ¬ m_manifold R Tdict.
The rest of this subproof is missing.
Proof:
Let L and TL be given.
Assume HL HTL.
We will prove locally_m_euclidean L TL (Sing Empty) normal_space L TL ¬ metrizable L TL.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hloc.
The rest of this subproof is missing.
Proof:
Let X, Tx and m be given.
Assume Hloc.
We will prove metrizable X Tx (paracompact_space X Tx Hausdorff_space X Tx).
The rest of this subproof is missing.
Theorem. (supp_ex_locally_euclidean_9)
∀X Tx m : set, locally_m_euclidean X Tx mmetrizable X Tx∀C : set, C Xm_manifold C (subspace_topology X Tx C)
Proof:
Let X, Tx and m be given.
Assume Hloc Hmet.
Let C be given.
Assume HC.
We will prove m_manifold C (subspace_topology X Tx C).
The rest of this subproof is missing.
Definition. We define Gdelta_in to be λX Tx A ⇒ ∃Fam : set, countable Fam (∀UFam, open_in X Tx U) Intersection_Fam Fam = A of type setsetsetprop.
Definition. We define open_map to be λX Tx Y Ty f ⇒ topology_on X Tx topology_on Y Ty function_on f X Y ∀U : set, U Txapply_fun f U Ty of type setsetsetsetsetprop.
Definition. We define topological_group to be λG Tg ⇒ topology_on G Tg ∃mult inv e : set, function_on mult (OrderedPair G G) G function_on inv G G e G continuous_map (OrderedPair G G) (product_topology G Tg G Tg) G Tg mult continuous_map G Tg G Tg inv of type setsetprop.
Definition. We define separated_subsets to be λX Tx A B ⇒ closure_of X Tx A B = Empty A closure_of X Tx B = Empty of type setsetsetsetprop.
Definition. We define completely_normal_space to be λX Tx ⇒ normal_space X Tx (∀A B : set, separated_subsets X Tx A B∃U V : set, open_in X Tx U open_in X Tx V A U B V U V = Empty) of type setsetprop.
Definition. We define linear_continuum to be λX Tx ⇒ ∃less : setsetprop, Tx = order_topology X (∀A : set, A XA Empty(∃upper : set, upper X ∀a : set, a Aless a upper)∃lub : set, lub X (∀a : set, a Aless a lub a = lub) (∀bound : set, bound X(∀a : set, a Aless a bound a = bound)less lub bound lub = bound)) of type setsetprop.
Proof:
Let X, Tx and x be given.
Assume H1: first_countable_space X Tx.
Assume H2: T1_space X Tx.
Assume H3: x X.
We will prove Gdelta_in X Tx (Sing x).
The rest of this subproof is missing.
Proof:
We will prove ∃X : set, ∃Tx : set, topology_on X Tx (∀x : set, x XGdelta_in X Tx (Sing x)) ¬ first_countable_space X Tx.
The rest of this subproof is missing.
Theorem. (ex30_2_basis_contains_countable)
∀X Tx : set, ∀Basis : set, second_countable_space X Txbasis_on X Basis∃CountableSub : set, CountableSub Basis countable CountableSub basis_on X CountableSub
Proof:
Let X, Tx and Basis be given.
Assume H1: second_countable_space X Tx.
Assume H2: basis_on X Basis.
We will prove ∃CountableSub : set, CountableSub Basis countable CountableSub basis_on X CountableSub.
The rest of this subproof is missing.
Proof:
Let X, Tx and A be given.
Assume H1: second_countable_space X Tx.
Assume H2: A X.
Assume H3: ¬ countable A.
We will prove ¬ countable {xA|limit_point_of x A X Tx}.
The rest of this subproof is missing.
Proof:
Let X, Tx and d be given.
Assume H1: compact_space X Tx.
Assume H2: metrizable X Tx.
Assume H3: metric_on X d.
Assume H4: Tx = metric_topology X d.
We will prove second_countable_space X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hmet: metrizable X Tx.
Assume Hdense: ∃D : set, D X countable D dense_in D X Tx.
We will prove second_countable_space X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hmet: metrizable X Tx.
Assume Hlin: Lindelof_space X Tx.
We will prove second_countable_space X Tx.
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
Let Tx be given.
Assume H: Tx = R.
We will prove ¬ metrizable ordered_square Tx.
The rest of this subproof is missing.
Proof:
Let Tx_SO and Tx_SbarO be given.
Assume H1: Tx_SO = SOmega_topology.
Assume H2: Tx_SbarO = SbarOmega_topology.
The rest of this subproof is missing.
Proof:
Let Tx be given.
Assume H: Tx = R.
The rest of this subproof is missing.
Proof:
Let X, Tx and A be given.
Assume Hlin: Lindelof_space X Tx.
Assume Hcl: closed_in X Tx A.
We will prove Lindelof_space A (subspace_topology X Tx A).
The rest of this subproof is missing.
Theorem. (ex30_9b_dense_not_countable_dense)
∃X : set, ∃Tx : set, ∃A : set, (∃D : set, D X countable D dense_in D X Tx) dense_in A X Tx ¬ (∃DA : set, DA A countable DA dense_in DA A (subspace_topology X Tx A))
Proof:
We will prove ∃X : set, ∃Tx : set, ∃A : set, (∃D : set, D X countable D dense_in D X Tx) dense_in A X Tx ¬ (∃DA : set, DA A countable DA dense_in DA A (subspace_topology X Tx A)).
The rest of this subproof is missing.
Theorem. (ex30_10_product_countable_dense)
∀Idx : set, ∀Fam : set, countable Idx(∀i : set, i Idx∃Xi : set, ∃Txi : set, ∃Di : set, apply_fun Fam i = OrderedPair Xi Txi Di Xi countable Di dense_in Di Xi Txi)∃D : set, D product_space Idx Fam countable D dense_in D (product_space Idx Fam) (product_topology_full Idx Fam)
Proof:
Let Idx and Fam be given.
Assume H1: countable Idx.
Assume H2: ∀i : set, i Idx∃Xi : set, ∃Txi : set, ∃Di : set, apply_fun Fam i = OrderedPair Xi Txi Di Xi countable Di dense_in Di Xi Txi.
We will prove ∃D : set, D product_space Idx Fam countable D dense_in D (product_space Idx Fam) (product_topology_full Idx Fam).
The rest of this subproof is missing.
Proof:
Let X, Tx, Y, Ty and f be given.
Assume H1: Lindelof_space X Tx.
Assume H2: continuous_map X Tx Y Ty f.
We will prove Lindelof_space (apply_fun f X) (subspace_topology Y Ty (apply_fun f X)).
The rest of this subproof is missing.
Theorem. (ex30_11b_image_countable_dense)
∀X Tx Y Ty f : set, (∃D : set, D X countable D dense_in D X Tx)continuous_map X Tx Y Ty f∃Df : set, Df (apply_fun f X) countable Df dense_in Df (apply_fun f X) (subspace_topology Y Ty (apply_fun f X))
Proof:
Let X, Tx, Y, Ty and f be given.
Assume H1: ∃D : set, D X countable D dense_in D X Tx.
Assume H2: continuous_map X Tx Y Ty f.
We will prove ∃Df : set, Df (apply_fun f X) countable Df dense_in Df (apply_fun f X) (subspace_topology Y Ty (apply_fun f X)).
The rest of this subproof is missing.
Proof:
Let X, Tx, Y, Ty and f be given.
Assume H1: first_countable_space X Tx.
Assume H2: continuous_map X Tx Y Ty f.
Assume H3: open_map X Tx Y Ty f.
The rest of this subproof is missing.
Proof:
Let X, Tx, Y, Ty and f be given.
Assume H1: second_countable_space X Tx.
Assume H2: continuous_map X Tx Y Ty f.
Assume H3: open_map X Tx Y Ty f.
The rest of this subproof is missing.
Theorem. (ex30_13_disjoint_open_sets_countable)
∀X Tx : set, (∃D : set, D X countable D dense_in D X Tx)∀Fam : set, (∀U : set, U Famopen_in X Tx U)(∀U V : set, U FamV FamU VU V = Empty)countable Fam
Proof:
Let X and Tx be given.
Assume H1: ∃D : set, D X countable D dense_in D X Tx.
Let Fam be given.
Assume H2: ∀U : set, U Famopen_in X Tx U.
Assume H3: ∀U V : set, U FamV FamU VU V = Empty.
We will prove countable Fam.
The rest of this subproof is missing.
Proof:
Let X, Tx, Y, Ty, Idx and Fam be given.
Assume H1: Lindelof_space X Tx.
Assume H2: compact_space Y Ty.
We will prove Lindelof_space (product_space Idx Fam) (product_topology_full Idx Fam).
The rest of this subproof is missing.
Proof:
We will prove ∃CI : set, ∃TCI : set, ∃D : set, D CI countable D dense_in D CI TCI second_countable_space CI TCI.
The rest of this subproof is missing.
Proof:
We will prove ∃Idx : set, ∃Fam : set, ∃D : set, D product_space Idx Fam countable D dense_in D (product_space Idx Fam) (product_topology_full Idx Fam).
The rest of this subproof is missing.
Proof:
Let J be given.
Assume H1: atleastp (𝒫 ω) J.
Assume H2: ¬ equip J (𝒫 ω).
Let Fam be given.
We will prove ¬ (∃D : set, D product_space J Fam countable D dense_in D (product_space J Fam) (product_topology_full J Fam)).
The rest of this subproof is missing.
Theorem. (ex30_17_Romega_box_countability)
∃Romega : set, ∃BoxTop : set, ∃Qinf : set, ∃SubTop : set, SubTop = subspace_topology Romega BoxTop Qinf (first_countable_space Qinf SubTop ¬ first_countable_space Qinf SubTop) (second_countable_space Qinf SubTop ¬ second_countable_space Qinf SubTop) (Lindelof_space Qinf SubTop ¬ Lindelof_space Qinf SubTop) ((∃D : set, D Qinf countable D dense_in D Qinf SubTop) ¬ (∃D : set, D Qinf countable D dense_in D Qinf SubTop))
Proof:
We will prove ∃Romega : set, ∃BoxTop : set, ∃Qinf : set, ∃SubTop : set, SubTop = subspace_topology Romega BoxTop Qinf (first_countable_space Qinf SubTop ¬ first_countable_space Qinf SubTop) (second_countable_space Qinf SubTop ¬ second_countable_space Qinf SubTop) (Lindelof_space Qinf SubTop ¬ Lindelof_space Qinf SubTop) ((∃D : set, D Qinf countable D dense_in D Qinf SubTop) ¬ (∃D : set, D Qinf countable D dense_in D Qinf SubTop)).
The rest of this subproof is missing.
Proof:
Let G and Tg be given.
Assume H1: topological_group G Tg.
Assume H2: first_countable_space G Tg.
Assume H3: (∃D : set, D G countable D dense_in D G Tg) Lindelof_space G Tg.
We will prove second_countable_space G Tg.
The rest of this subproof is missing.
Theorem. (ex31_1_regular_disjoint_closure_neighborhoods)
∀X Tx x y : set, regular_space X Txx Xy Xx y∃U V : set, open_in X Tx U open_in X Tx V x U y V closure_of X Tx U closure_of X Tx V = Empty
Proof:
Let X, Tx, x and y be given.
Assume Hreg: regular_space X Tx.
Assume Hx: x X.
Assume Hy: y X.
Assume Hneq: x y.
We will prove ∃U V : set, open_in X Tx U open_in X Tx V x U y V closure_of X Tx U closure_of X Tx V = Empty.
The rest of this subproof is missing.
Theorem. (ex31_2_normal_disjoint_closure_neighborhoods)
∀X Tx A B : set, normal_space X Txclosed_in X Tx Aclosed_in X Tx BA B = Empty∃U V : set, open_in X Tx U open_in X Tx V A U B V closure_of X Tx U closure_of X Tx V = Empty
Proof:
Let X, Tx, A and B be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume HB: closed_in X Tx B.
Assume Hdisj: A B = Empty.
We will prove ∃U V : set, open_in X Tx U open_in X Tx V A U B V closure_of X Tx U closure_of X Tx V = Empty.
The rest of this subproof is missing.
Proof:
Let X be given.
We will prove regular_space X (order_topology X).
The rest of this subproof is missing.
Theorem. (ex31_4_comparison_topologies_separation)
∀X Tx Tx' : set, Tx Tx'((Hausdorff_space X Tx'Hausdorff_space X Tx) (regular_space X Txregular_space X Tx') (normal_space X Txnormal_space X Tx'))
Proof:
Let X, Tx and Tx' be given.
Assume Hfiner: Tx Tx'.
Apply and3I to the current goal.
We will prove Hausdorff_space X Tx'Hausdorff_space X Tx.
The rest of this subproof is missing.
We will prove regular_space X Txregular_space X Tx'.
The rest of this subproof is missing.
We will prove normal_space X Txnormal_space X Tx'.
The rest of this subproof is missing.
Theorem. (ex31_5_equalizer_closed_in_Hausdorff)
∀X Tx Y Ty f g : set, continuous_map X Tx Y Ty fcontinuous_map X Tx Y Ty gHausdorff_space Y Tyclosed_in X Tx {xX|apply_fun f x = apply_fun g x}
Proof:
Let X, Tx, Y, Ty, f and g be given.
Assume Hf: continuous_map X Tx Y Ty f.
Assume Hg: continuous_map X Tx Y Ty g.
Assume HHaus: Hausdorff_space Y Ty.
We will prove closed_in X Tx {xX|apply_fun f x = apply_fun g x}.
The rest of this subproof is missing.
Theorem. (ex31_6_closed_map_preserves_normal)
∀X Tx Y Ty p : set, normal_space X Txcontinuous_map X Tx Y Ty p(∀A : set, closed_in X Tx Aclosed_in Y Ty (apply_fun p A))(∀y : set, y Y∃x : set, x X apply_fun p x = y)normal_space Y Ty
Proof:
Let X, Tx, Y, Ty and p be given.
Assume Hnorm: normal_space X Tx.
Assume Hcont: continuous_map X Tx Y Ty p.
Assume Hclosed: ∀A : set, closed_in X Tx Aclosed_in Y Ty (apply_fun p A).
Assume Hsurj: ∀y : set, y Y∃x : set, x X apply_fun p x = y.
We will prove normal_space Y Ty.
The rest of this subproof is missing.
Theorem. (ex31_7_perfect_map_properties)
∀X Tx Y Ty p : set, continuous_map X Tx Y Ty p(∀A : set, closed_in X Tx Aclosed_in Y Ty (apply_fun p A))(∀y : set, y Y∃x : set, x X apply_fun p x = y)(∀y : set, y Ycompact_space {xX|apply_fun p x = y} (subspace_topology X Tx {xX|apply_fun p x = y}))(Hausdorff_space X TxHausdorff_space Y Ty) (regular_space X Txregular_space Y Ty) (locally_compact X Txlocally_compact Y Ty) (second_countable_space X Txsecond_countable_space Y Ty)
Proof:
Let X, Tx, Y, Ty and p be given.
Assume Hcont: continuous_map X Tx Y Ty p.
Assume Hclosed: ∀A : set, closed_in X Tx Aclosed_in Y Ty (apply_fun p A).
Assume Hsurj: ∀y : set, y Y∃x : set, x X apply_fun p x = y.
Assume Hcompact: ∀y : set, y Ycompact_space {xX|apply_fun p x = y} (subspace_topology X Tx {xX|apply_fun p x = y}).
The rest of this subproof is missing.
Theorem. (ex31_8_orbit_space_properties)
∀G Tg X Tx alpha : set, topological_group G Tgcompact_space G Tg(Hausdorff_space X Tx∃XG TxG : set, Hausdorff_space XG TxG) (regular_space X Tx∃XG TxG : set, regular_space XG TxG) (normal_space X Tx∃XG TxG : set, normal_space XG TxG) (locally_compact X Tx∃XG TxG : set, locally_compact XG TxG) (second_countable_space X Tx∃XG TxG : set, second_countable_space XG TxG)
Proof:
Let G, Tg, X, Tx and alpha be given.
Assume Hgrp: topological_group G Tg.
Assume Hcomp: compact_space G Tg.
We will prove (Hausdorff_space X Tx∃XG TxG : set, Hausdorff_space XG TxG) (regular_space X Tx∃XG TxG : set, regular_space XG TxG) (normal_space X Tx∃XG TxG : set, normal_space XG TxG) (locally_compact X Tx∃XG TxG : set, locally_compact XG TxG) (second_countable_space X Tx∃XG TxG : set, second_countable_space XG TxG).
The rest of this subproof is missing.
Theorem. (ex31_9_Sorgenfrey_plane_no_separation)
∃Rl2 Tl2 A B : set, ¬ (∃U V : set, open_in Rl2 Tl2 U open_in Rl2 Tl2 V A U B V U V = Empty)
Proof:
We will prove ∃Rl2 Tl2 A B : set, ¬ (∃U V : set, open_in Rl2 Tl2 U open_in Rl2 Tl2 V A U B V U V = Empty).
The rest of this subproof is missing.
Proof:
Let X, Tx and A be given.
Assume Hnorm: normal_space X Tx.
Assume Hcl: closed_in X Tx A.
We will prove normal_space A (subspace_topology X Tx A).
The rest of this subproof is missing.
Theorem. (ex32_2_factors_inherit_separation)
∀Idx Fam : set, (∀i : set, i Idx∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi Xi Empty)((Hausdorff_space (product_space Idx Fam) (product_topology_full Idx Fam)∀i : set, i Idx∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi Hausdorff_space Xi Txi) (regular_space (product_space Idx Fam) (product_topology_full Idx Fam)∀i : set, i Idx∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi regular_space Xi Txi) (normal_space (product_space Idx Fam) (product_topology_full Idx Fam)∀i : set, i Idx∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi normal_space Xi Txi))
Proof:
Let Idx and Fam be given.
Assume Hnemp: ∀i : set, i Idx∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi Xi Empty.
We will prove (Hausdorff_space (product_space Idx Fam) (product_topology_full Idx Fam)∀i : set, i Idx∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi Hausdorff_space Xi Txi) (regular_space (product_space Idx Fam) (product_topology_full Idx Fam)∀i : set, i Idx∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi regular_space Xi Txi) (normal_space (product_space Idx Fam) (product_topology_full Idx Fam)∀i : set, i Idx∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi normal_space Xi Txi).
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume Hh: Hausdorff_space X Tx.
We will prove regular_space X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hlin: Lindelof_space X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ex32_6_completely_normal_characterization)
∀X Tx : set, completely_normal_space X Tx (∀A B : set, separated_subsets X Tx A B∃U V : set, open_in X Tx U open_in X Tx V A U B V U V = Empty)
Proof:
Let X and Tx be given.
Apply iffI to the current goal.
Assume H1: completely_normal_space X Tx.
Let A and B be given.
Assume H2: separated_subsets X Tx A B.
Apply H1 to the current goal.
Assume Hnorm Hsep.
An exact proof term for the current goal is Hsep A B H2.
Assume H1: ∀A B : set, separated_subsets X Tx A B∃U V : set, open_in X Tx U open_in X Tx V A U B V U V = Empty.
We will prove completely_normal_space X Tx.
We will prove normal_space X Tx (∀A B : set, separated_subsets X Tx A B∃U V : set, open_in X Tx U open_in X Tx V A U B V U V = Empty).
Apply andI to the current goal.
We will prove normal_space X Tx.
The rest of this subproof is missing.
We will prove ∀A B : set, separated_subsets X Tx A B∃U V : set, open_in X Tx U open_in X Tx V A U B V U V = Empty.
An exact proof term for the current goal is H1.
Proof:
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hlc: linear_continuum X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
Proof:
Let J be given.
Assume Huncnt: ¬ countable J.
The rest of this subproof is missing.
Definition. We define perfectly_normal_space to be λX Tx ⇒ normal_space X Tx (∀A : set, closed_in X Tx AGdelta_in X Tx A) of type setsetprop.
Theorem. (ex33_1_level_sets_urysohn)
∀X Tx A B : set, ∀U : setset, normal_space X Txclosed_in X Tx Aclosed_in X Tx BA B = Empty∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1) (∀r : set, apply_fun f X = {xX|apply_fun f x = r})
Proof:
Let X, Tx, A, B and U be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume HB: closed_in X Tx B.
Assume Hdisj: A B = Empty.
We will prove ∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1) (∀r : set, apply_fun f X = {xX|apply_fun f x = r}).
The rest of this subproof is missing.
Theorem. (ex33_2a_connected_normal_uncountable)
∀X Tx : set, connected_space X Txnormal_space X Tx(∃x y : set, x X y X x y)¬ countable X
Proof:
Let X and Tx be given.
Assume Hconn: connected_space X Tx.
Assume Hnorm: normal_space X Tx.
Assume Hneq: ∃x y : set, x X y X x y.
We will prove ¬ countable X.
The rest of this subproof is missing.
Theorem. (ex33_2b_connected_regular_uncountable)
∀X Tx : set, connected_space X Txregular_space X Tx(∃x y : set, x X y X x y)¬ countable X
Proof:
Let X and Tx be given.
Assume Hconn: connected_space X Tx.
Assume Hreg: regular_space X Tx.
Assume Hneq: ∃x y : set, x X y X x y.
We will prove ¬ countable X.
The rest of this subproof is missing.
Theorem. (ex33_3_urysohn_metric_direct)
∀X d A B : set, metric_on X dclosed_in X (metric_topology X d) Aclosed_in X (metric_topology X d) BA B = Empty∃f : set, continuous_map X (metric_topology X d) R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1)
Proof:
Let X, d, A and B be given.
Assume Hmet: metric_on X d.
Assume HcA: closed_in X (metric_topology X d) A.
Assume HcB: closed_in X (metric_topology X d) B.
Assume Hdisj: A B = Empty.
We will prove ∃f : set, continuous_map X (metric_topology X d) R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1).
The rest of this subproof is missing.
Theorem. (ex33_4_closed_Gdelta_vanishing_function)
∀X Tx A : set, normal_space X Txclosed_in X Tx A(Gdelta_in X Tx A ∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x A¬ (apply_fun f x = 0)))
Proof:
Let X, Tx and A be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Apply iffI to the current goal.
Assume HG: Gdelta_in X Tx A.
We will prove ∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x A¬ (apply_fun f x = 0)).
The rest of this subproof is missing.
Assume Hf: ∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x A¬ (apply_fun f x = 0)).
We will prove Gdelta_in X Tx A.
The rest of this subproof is missing.
Theorem. (ex33_5_strong_urysohn)
∀X Tx A B : set, normal_space X Txclosed_in X Tx Aclosed_in X Tx BA B = Empty(Gdelta_in X Tx A Gdelta_in X Tx B ∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1) (∀x : set, x Xx Ax B¬ (apply_fun f x = 0) ¬ (apply_fun f x = 1)))
Proof:
Let X, Tx, A and B be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume HB: closed_in X Tx B.
Assume Hdisj: A B = Empty.
Apply iffI to the current goal.
Assume HG: Gdelta_in X Tx A Gdelta_in X Tx B.
We will prove ∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1) (∀x : set, x Xx Ax B¬ (apply_fun f x = 0) ¬ (apply_fun f x = 1)).
The rest of this subproof is missing.
Assume Hf: ∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1) (∀x : set, x Xx Ax B¬ (apply_fun f x = 0) ¬ (apply_fun f x = 1)).
We will prove Gdelta_in X Tx A Gdelta_in X Tx B.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hmet: metrizable X Tx.
We will prove perfectly_normal_space X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hperf: perfectly_normal_space X Tx.
We will prove completely_normal_space X Tx.
The rest of this subproof is missing.
Proof:
We will prove ∃X Tx : set, completely_normal_space X Tx ¬ perfectly_normal_space X Tx.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume Hh: Hausdorff_space X Tx.
We will prove completely_regular_space X Tx.
The rest of this subproof is missing.
Theorem. (ex33_8_compact_subset_continuous_separation)
∀X Tx A B : set, completely_regular_space X Txcompact_space A (subspace_topology X Tx A)closed_in X Tx BA B = Empty∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1)
Proof:
Let X, Tx, A and B be given.
Assume Hcr: completely_regular_space X Tx.
Assume Hcpt: compact_space A (subspace_topology X Tx A).
Assume HcB: closed_in X Tx B.
Assume Hdisj: A B = Empty.
We will prove ∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1).
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
Let G and Tg be given.
Assume Htg: topological_group G Tg.
We will prove completely_regular_space G Tg.
The rest of this subproof is missing.
Proof:
We will prove ∃X Tx : set, regular_space X Tx ¬ completely_regular_space X Tx.
The rest of this subproof is missing.
Definition. We define locally_metrizable_space to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∃N : set, N Tx x N ∃d : set, metric_on N d subspace_topology X Tx N = metric_topology N d of type setsetprop.
Definition. We define retraction_of to be λX Tx A ⇒ A X ∃r : set, function_on r X X continuous_map X Tx X Tx r (∀x : set, x Xapply_fun r x A) (∀x : set, x Aapply_fun r x = x) of type setsetsetprop.
Definition. We define image_of_map to be λX Tx Y Ty f ⇒ {apply_fun f x|xX} of type setsetsetsetsetset.
Definition. We define absolute_retract to be λX Tx ⇒ Hausdorff_space X Tx ∀Y Ty, normal_space Y Ty∃e : set, embedding_of X Tx Y Ty e ∃r : set, retraction_of Y Ty (image_of_map X Tx Y Ty e) of type setsetprop.
Definition. We define coherent_topology to be λX Tx Y Ty ⇒ topology_on X Tx topology_on Y Ty X Y subspace_topology Y Ty X = Tx of type setsetsetsetprop.
Definition. We define compact_spaces_family to be λI Xi ⇒ ∀i : set, i Icompact_space (product_component Xi i) (product_component_topology Xi i) of type setsetprop.
Definition. We define surjective_map to be λX Y f ⇒ function_on f X Y ∀y : set, y Y∃x : set, x X apply_fun f x = y of type setsetsetprop.
Definition. We define ex34_1_Hausdorff_countable_basis_not_metrizable_example to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx Hausdorff_space X Tx second_countable_space X Tx ¬ metrizable X Tx} of type set.
Definition. We define ex34_2_completely_normal_not_metrizable_example to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx completely_normal_space X Tx ¬ metrizable X Tx} of type set.
Definition. We define ex34_3_compact_Hausdorff_metrizable_iff_second_countable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx compact_space X Tx Hausdorff_space X Tx (metrizable X Tx second_countable_space X Tx)} of type set.
Definition. We define ex34_4_locally_compact_Hausdorff_metrizable_questions to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx locally_compact X Tx Hausdorff_space X Tx (second_countable_space X Txmetrizable X Tx)} of type set.
Definition. We define ex34_5_one_point_compactification_metrizable_questions to be {q𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty p : set, q = OrderedPair (OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty)) p one_point_compactification X Tx Y Ty p Y ¬ p X (metrizable X Tx metrizable Y Ty)} of type set.
Definition. We define ex34_6_check_imbedding_proof to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx f : set, p = OrderedPair (OrderedPair X Tx) f completely_regular_space X Tx Hausdorff_space X Tx embedding_of X Tx (power_real ω) (product_topology_full ω (const_family ω R)) f} of type set.
Definition. We define ex34_7_locally_metrizable_compact_Hausdorff_metrizable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx locally_metrizable_space X Tx compact_space X Tx Hausdorff_space X Tx metrizable X Tx} of type set.
Definition. We define ex34_8_regular_Lindelof_locally_metrizable_metrizable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx (regular_space X Tx Lindelof_space X Tx locally_metrizable_space X Txmetrizable X Tx)} of type set.
Definition. We define ex34_9_compact_union_two_metrizable_closed_metrizable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A B : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair A B) compact_space X Tx Hausdorff_space X Tx closed_in X Tx A closed_in X Tx B (UPair A B) = X metrizable A (subspace_topology X Tx A) metrizable B (subspace_topology X Tx B) metrizable X Tx} of type set.
Definition. We define ex35_1_Tietze_implies_Urysohn to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx normal_space X Tx (∀A B : set, closed_in X Tx A closed_in X Tx B A B = Empty∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1))} of type set.
Definition. We define ex35_2_interval_partition_parameter to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx normal_space X Tx} of type set.
Definition. We define ex35_3_boundedness_equivalences_metrizable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx d : set, p = OrderedPair (OrderedPair X Tx) d metric_on X d metric_topology X d = Tx} of type set.
Definition. We define ex35_4_retract_properties to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A retraction_of X Tx A} of type set.
Definition. We define ex35_5_universal_extension_retracts to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A normal_space X Tx retraction_of X Tx A ∀Y Ty f : set, continuous_map A (subspace_topology X Tx A) Y Ty f∃g : set, continuous_map X Tx Y Ty g ∀x : set, x Aapply_fun g x = apply_fun f x} of type set.
Definition. We define ex35_6_absolute_retract_universal_extension to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx absolute_retract X Tx} of type set.
Definition. We define ex35_7_retract_examples to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A retraction_of X Tx A} of type set.
Definition. We define ex35_8_absolute_retract_equivalence to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx absolute_retract X Tx} of type set.
Definition. We define ex35_9_coherent_topology_normal to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) (topology_on X Tx topology_on Y Ty coherent_topology X Tx Y Tynormal_space Y Ty)} of type set.
Definition. We define ex36_manifold_embedding_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃M TM f : set, p = OrderedPair (OrderedPair M TM) f m_manifold M TM∃n : set, embedding_of M TM (euclidean_space n) (euclidean_topology n) f} of type set.
Definition. We define ex37_tychonoff_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃I Xi : set, p = OrderedPair I Xi compact_spaces_family I Xi compact_space (product_space I Xi) (product_topology_full I Xi)} of type set.
Definition. We define ex38_stone_cech_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) completely_regular_space X Tx compact_space Y Ty Hausdorff_space Y Ty ∃e : set, embedding_of X Tx Y Ty e} of type set.
Definition. We define ex39_local_finiteness_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx U : set, p = OrderedPair (OrderedPair X Tx) U locally_finite_family X Tx U} of type set.
Definition. We define ex40_nagata_smirnov_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx B : set, p = OrderedPair (OrderedPair X Tx) B (regular_space X Tx basis_on X B locally_finite_family X Tx Bmetrizable X Tx)} of type set.
Definition. We define ex41_paracompactness_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx U : set, p = OrderedPair (OrderedPair X Tx) U paracompact_space X Tx open_cover X Tx U} of type set.
Definition. We define ex42_smirnov_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx B : set, p = OrderedPair (OrderedPair X Tx) B (regular_space X Tx basis_on X B locally_finite_family X Tx Bmetrizable X Tx)} of type set.
Definition. We define ex43_complete_metric_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d Tx : set, p = OrderedPair (OrderedPair X d) Tx metric_on X d Tx = metric_topology X d complete_metric_space X d} of type set.
Definition. We define ex44_space_filling_exercises to be {f𝒫 (𝒫 (𝒫 R))|continuous_map unit_interval R2_standard_topology unit_square unit_square_topology f surjective_map unit_interval unit_square f} of type set.
Definition. We define ex45_compact_metric_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d Tx : set, p = OrderedPair (OrderedPair X d) Tx metric_on X d Tx = metric_topology X d compact_space X Tx} of type set.
Definition. We define ex46_convergence_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) topology_on X Tx topology_on Y Ty True} of type set.
Definition. We define ex47_ascoli_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) compact_space X Tx Hausdorff_space Y Ty} of type set.
Theorem. (ex48_1_Baire_union_interior)
∀X Tx : set, ∀Fam : set, Baire_space Txtopology_on X TxX Emptycountable_set FamX = Fam∃B : set, B Fam ∃U : set, U Tx U Empty U (closure_of X Tx B)
Proof:
Let X, Tx and Fam be given.
Assume HBaire: Baire_space Tx.
Assume Htop: topology_on X Tx.
Assume Hnemp: X Empty.
Assume Hcount: countable_set Fam.
Assume Hunion: X = Fam.
We will prove ∃B : set, B Fam ∃U : set, U Tx U Empty U (closure_of X Tx B).
The rest of this subproof is missing.
Theorem. (ex48_2_R_not_countable_empty_interior)
∀Fam : set, countable_set Fam(∀C : set, C Famclosed_in R R_standard_topology C (∀U : set, U R_standard_topologyU CU = Empty))R Fam
Proof:
Let Fam be given.
Assume Hcount: countable_set Fam.
Assume Hnowhere: ∀C : set, C Famclosed_in R R_standard_topology C (∀U : set, U R_standard_topologyU CU = Empty).
We will prove R Fam.
The rest of this subproof is missing.
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume HHaus: Hausdorff_space X Tx.
We will prove Baire_space Tx.
The rest of this subproof is missing.
Theorem. (ex48_4_locally_Baire_implies_Baire)
∀X Tx : set, topology_on X Tx(∀x : set, x X∃U : set, U Tx x U Baire_space (subspace_topology X Tx U))Baire_space Tx
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
Assume Hlocal: ∀x : set, x X∃U : set, U Tx x U Baire_space (subspace_topology X Tx U).
We will prove Baire_space Tx.
The rest of this subproof is missing.
Theorem. (ex48_5_Gdelta_Baire)
∀X Tx Y : set, (compact_space X Tx Hausdorff_space X Tx)(∃Fam : set, countable_set Fam (∀W : set, W FamW Tx) Y = intersection_over_family X Fam)Baire_space (subspace_topology X Tx Y)
Proof:
Let X, Tx and Y be given.
Assume Hcomp: compact_space X Tx Hausdorff_space X Tx.
Assume HGdelta: ∃Fam : set, countable_set Fam (∀W : set, W FamW Tx) Y = intersection_over_family X Fam.
We will prove Baire_space (subspace_topology X Tx Y).
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
Let f be given.
Assume Hf: function_on f R R.
We will prove ∃Fam : set, countable_set Fam (∀U : set, U FamU R_standard_topology) {xR|continuous_at f x} = intersection_over_family R Fam.
The rest of this subproof is missing.
Proof:
Let D be given.
Assume Hsub: D R.
Assume Hcount: countable_set D.
Assume Hdense: dense_in D R R_standard_topology.
We will prove ¬ (∃Fam : set, countable_set Fam (∀W : set, W FamW R_standard_topology) D = intersection_over_family R Fam).
The rest of this subproof is missing.
Theorem. (ex48_7_no_function_continuous_on_countable_dense)
∀D : set, D Rcountable_set Ddense_in D R R_standard_topology¬ (∃f : set, function_on f R R (∀x : set, x Dcontinuous_at f x) (∀x : set, x Rx D¬ continuous_at f x))
Proof:
Let D be given.
Assume Hsub: D R.
Assume Hcount: countable_set D.
Assume Hdense: dense_in D R R_standard_topology.
We will prove ¬ (∃f : set, function_on f R R (∀x : set, x Dcontinuous_at f x) (∀x : set, x Rx D¬ continuous_at f x)).
The rest of this subproof is missing.
Theorem. (ex48_8_pointwise_limit_continuity)
∀fn : set, ∀f : set, (∀n : set, n ωcontinuous_map R R_standard_topology R R_standard_topology (apply_fun fn n))function_on f R R(∀x : set, x R∃limval : set, limval R ∀eps : set, eps RTrue)¬ countable_set {xR|continuous_at f x}
Proof:
Let fn and f be given.
Assume Hf: function_on f R R.
Assume Hlim: ∀x : set, x R∃limval : set, limval R ∀eps : set, eps RTrue.
The rest of this subproof is missing.
Theorem. (ex48_9_Thomae_function)
∀g : set, ∀f : set, (∀n : set, n ωapply_fun g n Q)function_on f R R(∀n : set, n ωapply_fun f (apply_fun g n) = R)(∀x : set, x Rx Qapply_fun f x = Empty)∀x : set, x Rx Qcontinuous_at f x
Proof:
Let g and f be given.
Assume Hg: ∀n : set, n ωapply_fun g n Q.
Assume Hf: function_on f R R.
Assume Hfg: ∀n : set, n ωapply_fun f (apply_fun g n) = R.
Assume Hfirr: ∀x : set, x Rx Qapply_fun f x = Empty.
Let x be given.
Assume Hx: x R.
Assume Hirr: x Q.
We will prove continuous_at f x.
The rest of this subproof is missing.
Theorem. (ex48_10_uniform_boundedness)
∀X d : set, ∀FF : set, complete_metric_space X dFF 𝒫 (𝒫 R)(∀a : set, a X∃M : set, M R ∀f : set, f FFapply_fun f a R)∃U : set, ∃M : set, U (metric_topology X d) U Empty M R ∀f : set, f FF∀x : set, x Uapply_fun f x R
Proof:
Let X, d and FF be given.
Assume Hcomplete: complete_metric_space X d.
Assume HFF: FF 𝒫 (𝒫 R).
Assume Hbound: ∀a : set, a X∃M : set, M R ∀f : set, f FFapply_fun f a R.
We will prove ∃U : set, ∃M : set, U (metric_topology X d) U Empty M R ∀f : set, f FF∀x : set, x Uapply_fun f x R.
The rest of this subproof is missing.
Theorem. (ex48_11_Rl_Baire)
∀Tl : set, Tl = RBaire_space Tl
Proof:
Let Tl be given.
Assume HTl: Tl = R.
We will prove Baire_space Tl.
The rest of this subproof is missing.
Proof:
Let f, g and k be given.
Assume Hf: f = R.
Assume Hg: g = R.
Assume Hk: k = R.
The rest of this subproof is missing.
Proof:
Let n and eps be given.
Assume Hn: n ω.
Assume Heps: eps R.
The rest of this subproof is missing.
Definition. We define ex50_dimension_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx n : set, p = OrderedPair (OrderedPair X Tx) n topology_on X Tx ordinal n} of type set.